2cos (problem 3.3.5)

Percentage Accurate: 37.5% → 99.3%

Time: 17.4s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0028:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0034:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0028)
   (fma (sin eps) (- (sin x)) (* (cos x) (+ -1.0 (cos eps))))
   (if (<= eps 0.0034)
     (fma
      0.041666666666666664
      (* (cos x) (pow eps 4.0))
      (+
       (* -0.5 (* eps (* eps (cos x))))
       (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps))))
     (- (fma (cos x) (cos eps) (* (sin x) (- (sin eps)))) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0028) {
		tmp = fma(sin(eps), -sin(x), (cos(x) * (-1.0 + cos(eps))));
	} else if (eps <= 0.0034) {
		tmp = fma(0.041666666666666664, (cos(x) * pow(eps, 4.0)), ((-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps))));
	} else {
		tmp = fma(cos(x), cos(eps), (sin(x) * -sin(eps))) - cos(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0028)
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	elseif (eps <= 0.0034)
		tmp = fma(0.041666666666666664, Float64(cos(x) * (eps ^ 4.0)), Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps))));
	else
		tmp = Float64(fma(cos(x), cos(eps), Float64(sin(x) * Float64(-sin(eps)))) - cos(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0028], N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0034], N[(0.041666666666666664 * N[(N[Cos[x], $MachinePrecision] * N[Power[eps, 4.0], $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]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00279999999999999997

   1. Initial program 64.8%

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

      1. add-sqr-sqrt 29.8%

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

      2. sqrt-unprod 39.1%

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

      3. pow2 39.1%

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

   3. Applied egg-rr 39.1%

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

      1. sqrt-pow1 64.8%

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

      2. metadata-eval 64.8%

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

      3. cos-sum 99.0%

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

      4. pow1 99.0%

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

      5. fma-neg 98.9%

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

      6. *-commutative 98.9%

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

      7. distribute-rgt-neg-in 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in x around inf 99.0%

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

      1. associate--l+ 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 98.9%

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

      4. neg-mul-1 98.9%

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

      5. *-commutative 98.9%

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

      6. fma-def 99.0%

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

      7. *-rgt-identity 99.0%

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

      8. distribute-lft-out-- 99.0%

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

      9. sub-neg 99.0%

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

      10. metadata-eval 99.0%

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

      11. +-commutative 99.0%

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

   8. Simplified 99.0%

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

   if -0.00279999999999999997 < eps < 0.00339999999999999981

   1. Initial program 21.9%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. fma-def 99.8%

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

      2. *-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. unpow2 99.8%

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

      6. associate-*l* 99.8%

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

      7. associate-*r* 99.8%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. mul-1-neg 99.8%

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

   4. Simplified 99.8%

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

   if 0.00339999999999999981 < eps

   1. Initial program 51.7%

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

      1. cos-sum 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. fma-def 99.0%

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

   3. Applied egg-rr 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0028:\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0034:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664, \cos x \cdot {\varepsilon}^{4}, -0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \sin x \cdot \left(-\sin \varepsilon\right)\right) - \cos x\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-14} \lor \neg \left(x \leq 3 \cdot 10^{-59}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.9e-14) (not (<= x 3e-59)))
   (fma (sin eps) (- (sin x)) (* (cos x) (+ -1.0 (cos eps))))
   (* -2.0 (* (sin (* 0.5 (- eps (* x -2.0)))) (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.9e-14) || !(x <= 3e-59)) {
		tmp = fma(sin(eps), -sin(x), (cos(x) * (-1.0 + cos(eps))));
	} else {
		tmp = -2.0 * (sin((0.5 * (eps - (x * -2.0)))) * sin((eps * 0.5)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.9e-14) || !(x <= 3e-59))
		tmp = fma(sin(eps), Float64(-sin(x)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	else
		tmp = Float64(-2.0 * Float64(sin(Float64(0.5 * Float64(eps - Float64(x * -2.0)))) * sin(Float64(eps * 0.5))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.9e-14], N[Not[LessEqual[x, 3e-59]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] * (-N[Sin[x], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sin[N[(0.5 * N[(eps - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9000000000000001e-14 or 3.0000000000000001e-59 < x

   1. Initial program 14.9%

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

      1. add-sqr-sqrt 7.9%

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

      2. sqrt-unprod 12.5%

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

      3. pow2 12.5%

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

   3. Applied egg-rr 12.5%

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

      1. sqrt-pow1 14.9%

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

      2. metadata-eval 14.9%

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

      3. cos-sum 53.9%

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

      4. pow1 53.9%

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

      5. fma-neg 53.9%

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

      6. *-commutative 53.9%

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

      7. distribute-rgt-neg-in 53.9%

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

   5. Applied egg-rr 53.9%

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

   6. Taylor expanded in x around inf 53.9%

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

      1. associate--l+ 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. neg-mul-1 99.2%

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

      5. *-commutative 99.2%

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

      6. fma-def 99.2%

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

      7. *-rgt-identity 99.2%

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

      8. distribute-lft-out-- 99.2%

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

      9. sub-neg 99.2%

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

      10. metadata-eval 99.2%

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

      11. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   if -1.9000000000000001e-14 < x < 3.0000000000000001e-59

   1. Initial program 74.0%

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

      1. diff-cos 93.5%

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

      2. div-inv 93.5%

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

      3. metadata-eval 93.5%

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

      4. div-inv 93.5%

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

      5. +-commutative 93.5%

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

      6. metadata-eval 93.5%

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

   3. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

      2. +-commutative 93.5%

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

      3. associate--l+ 99.6%

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

      4. +-inverses 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around -inf 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-14} \lor \neg \left(x \leq 3 \cdot 10^{-59}\right):\\ \;\;\;\;\mathsf{fma}\left(\sin \varepsilon, -\sin x, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon - x \cdot -2\right)\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105 \lor \neg \left(\varepsilon \leq 0.0105\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0105) (not (<= eps 0.0105)))
   (- (cos eps) (cos x))
   (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0105) || !(eps <= 0.0105)) {
		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 <= (-0.0105d0)) .or. (.not. (eps <= 0.0105d0))) 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 <= -0.0105) || !(eps <= 0.0105)) {
		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 <= -0.0105) or not (eps <= 0.0105):
		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 <= -0.0105) || !(eps <= 0.0105))
		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 <= -0.0105) || ~((eps <= 0.0105)))
		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, -0.0105], N[Not[LessEqual[eps, 0.0105]], $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 < -0.0105000000000000007 or 0.0105000000000000007 < eps

   1. Initial program 58.6%

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

   2. Taylor expanded in x around 0 60.8%

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

   if -0.0105000000000000007 < eps < 0.0105000000000000007

   1. Initial program 21.9%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unsub-neg 99.3%

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

      3. unpow2 99.3%

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

      4. associate-*l* 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105 \lor \neg \left(\varepsilon \leq 0.0105\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 4: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

   1. diff-cos 49.2%

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

   2. div-inv 49.2%

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

   3. metadata-eval 49.2%

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

   4. div-inv 49.2%

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

   5. +-commutative 49.2%

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

   6. metadata-eval 49.2%

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

3. Applied egg-rr 49.2%

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

   1. *-commutative 49.2%

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

   2. +-commutative 49.2%

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

   3. associate--l+ 77.9%

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

   4. +-inverses 77.9%

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

   5. distribute-lft-in 77.9%

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

   6. metadata-eval 77.9%

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

   7. *-commutative 77.9%

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

   8. +-commutative 77.9%

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

5. Simplified 77.9%

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

6. Taylor expanded in x around -inf 77.9%

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

7. Final simplification 77.9%

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

Alternative 5: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

   1. diff-cos 49.2%

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

   2. div-inv 49.2%

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

   3. metadata-eval 49.2%

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

   4. div-inv 49.2%

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

   5. +-commutative 49.2%

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

   6. metadata-eval 49.2%

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

3. Applied egg-rr 49.2%

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

   1. *-commutative 49.2%

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

   2. +-commutative 49.2%

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

   3. associate--l+ 77.9%

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

   4. +-inverses 77.9%

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

   5. distribute-lft-in 77.9%

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

   6. metadata-eval 77.9%

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

   7. *-commutative 77.9%

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

   8. +-commutative 77.9%

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

5. Simplified 77.9%

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

6. Final simplification 77.9%

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

Alternative 6: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-16} \lor \neg \left(x \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;t_0 \cdot \left(\sin x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -1.2e-16) (not (<= x 1.8e-21)))
     (* t_0 (* (sin x) -2.0))
     (* -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 <= -1.2e-16) || !(x <= 1.8e-21)) {
		tmp = t_0 * (sin(x) * -2.0);
	} 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 <= (-1.2d-16)) .or. (.not. (x <= 1.8d-21))) then
        tmp = t_0 * (sin(x) * (-2.0d0))
    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 <= -1.2e-16) || !(x <= 1.8e-21)) {
		tmp = t_0 * (Math.sin(x) * -2.0);
	} 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 <= -1.2e-16) or not (x <= 1.8e-21):
		tmp = t_0 * (math.sin(x) * -2.0)
	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 <= -1.2e-16) || !(x <= 1.8e-21))
		tmp = Float64(t_0 * Float64(sin(x) * -2.0));
	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 <= -1.2e-16) || ~((x <= 1.8e-21)))
		tmp = t_0 * (sin(x) * -2.0);
	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, -1.2e-16], N[Not[LessEqual[x, 1.8e-21]], $MachinePrecision]], N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.20000000000000002e-16 or 1.79999999999999995e-21 < x

   1. Initial program 8.8%

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

      1. diff-cos 7.5%

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

      2. div-inv 7.5%

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

      3. metadata-eval 7.5%

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

      4. div-inv 7.5%

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

      5. +-commutative 7.5%

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

      6. metadata-eval 7.5%

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

   3. Applied egg-rr 7.5%

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

      1. *-commutative 7.5%

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

      2. +-commutative 7.5%

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

      3. associate--l+ 57.9%

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

      4. +-inverses 57.9%

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

      5. distribute-lft-in 57.9%

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

      6. metadata-eval 57.9%

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

      7. *-commutative 57.9%

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

      8. +-commutative 57.9%

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

   5. Simplified 57.9%

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

      1. expm1-log1p-u 57.9%

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

      2. *-commutative 57.9%

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

      3. +-commutative 57.9%

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

      4. +-rgt-identity 57.9%

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

   7. Applied egg-rr 57.9%

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

   8. Taylor expanded in eps around 0 56.6%

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

   9. Taylor expanded in x around inf 56.6%

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

       1. *-commutative 56.6%

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

       2. associate-*r* 56.6%

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

   11. Simplified 56.6%

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

   if -1.20000000000000002e-16 < x < 1.79999999999999995e-21

   1. Initial program 76.3%

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

      1. diff-cos 94.2%

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

      2. div-inv 94.2%

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

      3. metadata-eval 94.2%

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

      4. div-inv 94.2%

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

      5. +-commutative 94.2%

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

      6. metadata-eval 94.2%

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

   3. Applied egg-rr 94.2%

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

      1. *-commutative 94.2%

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

      2. +-commutative 94.2%

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

      3. associate--l+ 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 93.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-16} \lor \neg \left(x \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 7: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-15} \lor \neg \left(x \leq 8.5 \cdot 10^{-25}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.2e-15) (not (<= x 8.5e-25)))
   (* (sin x) (- eps))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.2e-15) || !(x <= 8.5e-25)) {
		tmp = sin(x) * -eps;
	} 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 <= (-3.2d-15)) .or. (.not. (x <= 8.5d-25))) then
        tmp = sin(x) * -eps
    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 <= -3.2e-15) || !(x <= 8.5e-25)) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.2e-15) or not (x <= 8.5e-25):
		tmp = math.sin(x) * -eps
	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 <= -3.2e-15) || !(x <= 8.5e-25))
		tmp = Float64(sin(x) * Float64(-eps));
	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 <= -3.2e-15) || ~((x <= 8.5e-25)))
		tmp = sin(x) * -eps;
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.2e-15], N[Not[LessEqual[x, 8.5e-25]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * (-eps)), $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 < -3.1999999999999999e-15 or 8.49999999999999981e-25 < x

   1. Initial program 8.8%

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

   2. Taylor expanded in eps around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. mul-1-neg 53.5%

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

   4. Simplified 53.5%

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

   if -3.1999999999999999e-15 < x < 8.49999999999999981e-25

   1. Initial program 76.3%

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

      1. diff-cos 94.2%

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

      2. div-inv 94.2%

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

      3. metadata-eval 94.2%

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

      4. div-inv 94.2%

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

      5. +-commutative 94.2%

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

      6. metadata-eval 94.2%

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

   3. Applied egg-rr 94.2%

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

      1. *-commutative 94.2%

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

      2. +-commutative 94.2%

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

      3. associate--l+ 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 93.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-15} \lor \neg \left(x \leq 8.5 \cdot 10^{-25}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 8: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.3 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 0.000114\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5.3e-8) (not (<= eps 0.000114)))
   (- (cos eps) (cos x))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.3e-8) || !(eps <= 0.000114)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-5.3d-8)) .or. (.not. (eps <= 0.000114d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.3e-8) || !(eps <= 0.000114)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5.3e-8) or not (eps <= 0.000114):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5.3e-8) || !(eps <= 0.000114))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5.3e-8) || ~((eps <= 0.000114)))
		tmp = cos(eps) - cos(x);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5.3e-8], N[Not[LessEqual[eps, 0.000114]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.2999999999999998e-8 or 1.1400000000000001e-4 < eps

   1. Initial program 58.5%

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

   2. Taylor expanded in x around 0 60.7%

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

   if -5.2999999999999998e-8 < eps < 1.1400000000000001e-4

   1. Initial program 21.6%

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

   2. Taylor expanded in eps around 0 81.7%

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

      1. associate-*r* 81.7%

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

      2. mul-1-neg 81.7%

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

   4. Simplified 81.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.3 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 0.000114\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 9: 48.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -7.9 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))) (t_1 (* eps (* eps -0.5))))
   (if (<= eps -0.00017)
     t_0
     (if (<= eps -7.9e-156)
       t_1
       (if (<= eps 3.1e-87) (* x (- eps)) (if (<= eps 0.00015) 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 <= -0.00017) {
		tmp = t_0;
	} else if (eps <= -7.9e-156) {
		tmp = t_1;
	} else if (eps <= 3.1e-87) {
		tmp = x * -eps;
	} else if (eps <= 0.00015) {
		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 <= (-0.00017d0)) then
        tmp = t_0
    else if (eps <= (-7.9d-156)) then
        tmp = t_1
    else if (eps <= 3.1d-87) then
        tmp = x * -eps
    else if (eps <= 0.00015d0) 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 <= -0.00017) {
		tmp = t_0;
	} else if (eps <= -7.9e-156) {
		tmp = t_1;
	} else if (eps <= 3.1e-87) {
		tmp = x * -eps;
	} else if (eps <= 0.00015) {
		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 <= -0.00017:
		tmp = t_0
	elif eps <= -7.9e-156:
		tmp = t_1
	elif eps <= 3.1e-87:
		tmp = x * -eps
	elif eps <= 0.00015:
		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 <= -0.00017)
		tmp = t_0;
	elseif (eps <= -7.9e-156)
		tmp = t_1;
	elseif (eps <= 3.1e-87)
		tmp = Float64(x * Float64(-eps));
	elseif (eps <= 0.00015)
		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 <= -0.00017)
		tmp = t_0;
	elseif (eps <= -7.9e-156)
		tmp = t_1;
	elseif (eps <= 3.1e-87)
		tmp = x * -eps;
	elseif (eps <= 0.00015)
		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, -0.00017], t$95$0, If[LessEqual[eps, -7.9e-156], t$95$1, If[LessEqual[eps, 3.1e-87], N[(x * (-eps)), $MachinePrecision], If[LessEqual[eps, 0.00015], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.7e-4 or 1.49999999999999987e-4 < eps

   1. Initial program 58.6%

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

   2. Taylor expanded in x around 0 58.7%

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

   if -1.7e-4 < eps < -7.89999999999999976e-156 or 3.09999999999999998e-87 < eps < 1.49999999999999987e-4

   1. Initial program 5.1%

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

   2. Taylor expanded in x around 0 5.4%

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

   3. Taylor expanded in eps around 0 46.8%

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

      1. *-commutative 46.8%

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

      2. unpow2 46.8%

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

      3. associate-*l* 46.8%

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

   5. Simplified 46.8%

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

   if -7.89999999999999976e-156 < eps < 3.09999999999999998e-87

   1. Initial program 31.4%

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

   2. Taylor expanded in x around 0 29.9%

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

   3. Taylor expanded in eps around 0 42.8%

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

      1. associate-*r* 42.8%

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

      2. mul-1-neg 42.8%

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

   5. Simplified 42.8%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -7.9 \cdot 10^{-156}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 10: 66.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 0.000114\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5.5e-8) (not (<= eps 0.000114)))
   (+ -1.0 (cos eps))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.5e-8) || !(eps <= 0.000114)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-5.5d-8)) .or. (.not. (eps <= 0.000114d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.5e-8) || !(eps <= 0.000114)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5.5e-8) or not (eps <= 0.000114):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5.5e-8) || !(eps <= 0.000114))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5.5e-8) || ~((eps <= 0.000114)))
		tmp = -1.0 + cos(eps);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5.5e-8], N[Not[LessEqual[eps, 0.000114]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.5000000000000003e-8 or 1.1400000000000001e-4 < eps

   1. Initial program 58.5%

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

   2. Taylor expanded in x around 0 58.6%

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

   if -5.5000000000000003e-8 < eps < 1.1400000000000001e-4

   1. Initial program 21.6%

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

   2. Taylor expanded in eps around 0 81.7%

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

      1. associate-*r* 81.7%

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

      2. mul-1-neg 81.7%

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

   4. Simplified 81.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 0.000114\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 11: 22.4% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-72}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.7e-72) (* eps (* eps -0.5)) (* x (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.7e-72) {
		tmp = eps * (eps * -0.5);
	} else {
		tmp = x * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.7d-72) then
        tmp = eps * (eps * (-0.5d0))
    else
        tmp = x * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.7e-72) {
		tmp = eps * (eps * -0.5);
	} else {
		tmp = x * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.7e-72:
		tmp = eps * (eps * -0.5)
	else:
		tmp = x * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.7e-72)
		tmp = Float64(eps * Float64(eps * -0.5));
	else
		tmp = Float64(x * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.7e-72)
		tmp = eps * (eps * -0.5);
	else
		tmp = x * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.7e-72], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], N[(x * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6999999999999999e-72

   1. Initial program 49.1%

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

   2. Taylor expanded in x around 0 49.1%

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

   3. Taylor expanded in eps around 0 27.2%

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

      1. *-commutative 27.2%

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

      2. unpow2 27.2%

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

      3. associate-*l* 27.2%

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

   5. Simplified 27.2%

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

   if 1.6999999999999999e-72 < x

   1. Initial program 23.7%

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

   2. Taylor expanded in x around 0 21.9%

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

   3. Taylor expanded in eps around 0 11.0%

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

      1. associate-*r* 11.0%

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

      2. mul-1-neg 11.0%

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

   5. Simplified 11.0%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-72}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 12: 17.4% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

2. Taylor expanded in x around 0 39.7%

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

3. Taylor expanded in eps around 0 16.2%

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

   1. associate-*r* 16.2%

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

   2. mul-1-neg 16.2%

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

5. Simplified 16.2%

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

6. Final simplification 16.2%

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

Alternative 13: 12.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 41.2%

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

2. Taylor expanded in x around 0 41.3%

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

3. Taylor expanded in eps around 0 11.8%

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

4. Final simplification 11.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))