2cos (problem 3.3.5)

Percentage Accurate: 38.6% → 98.7%

Time: 17.2s

Alternatives: 11

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: 38.6% 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: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := -\sin x\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \sin \varepsilon, \cos x \cdot \cos \varepsilon - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (- (sin x))))
   (if (<= x -2.6e-8)
     (fma t_1 (sin eps) (* (cos x) (+ (cos eps) -1.0)))
     (if (<= x 2.2e-82)
       (* -2.0 (* t_0 (+ t_0 (* x (cos (* eps 0.5))))))
       (fma t_1 (sin eps) (- (* (cos x) (cos eps)) (cos x)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = -sin(x);
	double tmp;
	if (x <= -2.6e-8) {
		tmp = fma(t_1, sin(eps), (cos(x) * (cos(eps) + -1.0)));
	} else if (x <= 2.2e-82) {
		tmp = -2.0 * (t_0 * (t_0 + (x * cos((eps * 0.5)))));
	} else {
		tmp = fma(t_1, sin(eps), ((cos(x) * cos(eps)) - cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(-sin(x))
	tmp = 0.0
	if (x <= -2.6e-8)
		tmp = fma(t_1, sin(eps), Float64(cos(x) * Float64(cos(eps) + -1.0)));
	elseif (x <= 2.2e-82)
		tmp = Float64(-2.0 * Float64(t_0 * Float64(t_0 + Float64(x * cos(Float64(eps * 0.5))))));
	else
		tmp = fma(t_1, sin(eps), Float64(Float64(cos(x) * cos(eps)) - cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[Sin[x], $MachinePrecision])}, If[LessEqual[x, -2.6e-8], N[(t$95$1 * N[Sin[eps], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e-82], N[(-2.0 * N[(t$95$0 * N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[eps], $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 x < -2.6000000000000001e-8

   1. Initial program 7.2%

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

      1. cos-sum 55.4%

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

      2. cancel-sign-sub-inv 55.4%

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

      3. fma-def 55.5%

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

   3. Applied egg-rr 55.5%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. *-commutative 55.4%

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

      2. neg-mul-1 55.4%

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

      3. associate--l+ 99.1%

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

      4. distribute-lft-neg-in 99.1%

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

      5. *-commutative 99.1%

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

      6. fma-def 99.2%

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

      7. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. sub-neg 99.2%

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

      2. neg-mul-1 99.2%

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

      3. distribute-rgt-out 99.3%

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

   9. Simplified 99.3%

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

   if -2.6000000000000001e-8 < x < 2.19999999999999986e-82

   1. Initial program 72.4%

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

      1. diff-cos 89.7%

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

      2. div-inv 89.7%

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

      3. metadata-eval 89.7%

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

      4. div-inv 89.7%

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

      5. +-commutative 89.7%

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

      6. metadata-eval 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. +-commutative 89.7%

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

      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. Step-by-step derivation

      1. add-log-exp 71.6%

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

      2. *-commutative 71.6%

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

      3. +-commutative 71.6%

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

      4. +-rgt-identity 71.6%

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

   7. Applied egg-rr 71.6%

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

   8. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. distribute-lft-out 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

   10. Simplified 99.5%

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

   if 2.19999999999999986e-82 < x

   1. Initial program 16.7%

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

      1. cos-sum 58.7%

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

      2. cancel-sign-sub-inv 58.7%

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

      3. fma-def 58.8%

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

   3. Applied egg-rr 58.8%

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

   4. Taylor expanded in x around inf 58.7%

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

      1. *-commutative 58.7%

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

      2. neg-mul-1 58.7%

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

      3. associate--l+ 98.3%

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

      4. distribute-lft-neg-in 98.3%

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

      5. *-commutative 98.3%

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

      6. fma-def 98.4%

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

      7. *-commutative 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \cos \varepsilon - \cos x\right)\\ \end{array} \]

Alternative 2: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 2.3 \cdot 10^{-82}\right):\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.7e-8) (not (<= x 2.3e-82)))
     (fma (- (sin x)) (sin eps) (* (cos x) (+ (cos eps) -1.0)))
     (* -2.0 (* t_0 (+ t_0 (* x (cos (* eps 0.5)))))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -2.7e-8) || !(x <= 2.3e-82)) {
		tmp = fma(-sin(x), sin(eps), (cos(x) * (cos(eps) + -1.0)));
	} else {
		tmp = -2.0 * (t_0 * (t_0 + (x * cos((eps * 0.5)))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -2.7e-8) || !(x <= 2.3e-82))
		tmp = fma(Float64(-sin(x)), sin(eps), Float64(cos(x) * Float64(cos(eps) + -1.0)));
	else
		tmp = Float64(-2.0 * Float64(t_0 * Float64(t_0 + Float64(x * cos(Float64(eps * 0.5))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.7e-8], N[Not[LessEqual[x, 2.3e-82]], $MachinePrecision]], N[((-N[Sin[x], $MachinePrecision]) * N[Sin[eps], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$0 * N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.70000000000000002e-8 or 2.29999999999999997e-82 < x

   1. Initial program 12.0%

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

      1. cos-sum 57.1%

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

      2. cancel-sign-sub-inv 57.1%

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

      3. fma-def 57.1%

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

   3. Applied egg-rr 57.1%

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

   4. Taylor expanded in x around inf 57.1%

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

      1. *-commutative 57.1%

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

      2. neg-mul-1 57.1%

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

      3. associate--l+ 98.7%

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

      4. distribute-lft-neg-in 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-def 98.8%

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

      7. *-commutative 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in x around inf 98.8%

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

      1. sub-neg 98.8%

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

      2. neg-mul-1 98.8%

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

      3. distribute-rgt-out 98.8%

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

   9. Simplified 98.8%

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

   if -2.70000000000000002e-8 < x < 2.29999999999999997e-82

   1. Initial program 72.4%

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

      1. diff-cos 89.7%

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

      2. div-inv 89.7%

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

      3. metadata-eval 89.7%

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

      4. div-inv 89.7%

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

      5. +-commutative 89.7%

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

      6. metadata-eval 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. +-commutative 89.7%

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

      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. Step-by-step derivation

      1. add-log-exp 71.6%

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

      2. *-commutative 71.6%

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

      3. +-commutative 71.6%

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

      4. +-rgt-identity 71.6%

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

   7. Applied egg-rr 71.6%

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

   8. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. distribute-lft-out 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-8} \lor \neg \left(x \leq 2.3 \cdot 10^{-82}\right):\\ \;\;\;\;\mathsf{fma}\left(-\sin x, \sin \varepsilon, \cos x \cdot \left(\cos \varepsilon + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{-8} \lor \neg \left(x \leq 1.8 \cdot 10^{-82}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(t_0 + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -2.85e-8) (not (<= x 1.8e-82)))
     (- (* (cos x) (+ (cos eps) -1.0)) (* (sin x) (sin eps)))
     (* -2.0 (* t_0 (+ t_0 (* x (cos (* eps 0.5)))))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -2.85e-8) || !(x <= 1.8e-82)) {
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(x) * sin(eps));
	} else {
		tmp = -2.0 * (t_0 * (t_0 + (x * cos((eps * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-2.85d-8)) .or. (.not. (x <= 1.8d-82))) then
        tmp = (cos(x) * (cos(eps) + (-1.0d0))) - (sin(x) * sin(eps))
    else
        tmp = (-2.0d0) * (t_0 * (t_0 + (x * cos((eps * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -2.85e-8) || !(x <= 1.8e-82)) {
		tmp = (Math.cos(x) * (Math.cos(eps) + -1.0)) - (Math.sin(x) * Math.sin(eps));
	} else {
		tmp = -2.0 * (t_0 * (t_0 + (x * Math.cos((eps * 0.5)))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -2.85e-8) or not (x <= 1.8e-82):
		tmp = (math.cos(x) * (math.cos(eps) + -1.0)) - (math.sin(x) * math.sin(eps))
	else:
		tmp = -2.0 * (t_0 * (t_0 + (x * math.cos((eps * 0.5)))))
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -2.85e-8) || !(x <= 1.8e-82))
		tmp = Float64(Float64(cos(x) * Float64(cos(eps) + -1.0)) - Float64(sin(x) * sin(eps)));
	else
		tmp = Float64(-2.0 * Float64(t_0 * Float64(t_0 + Float64(x * cos(Float64(eps * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -2.85e-8) || ~((x <= 1.8e-82)))
		tmp = (cos(x) * (cos(eps) + -1.0)) - (sin(x) * sin(eps));
	else
		tmp = -2.0 * (t_0 * (t_0 + (x * cos((eps * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -2.85e-8], N[Not[LessEqual[x, 1.8e-82]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$0 * N[(t$95$0 + N[(x * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85000000000000004e-8 or 1.79999999999999999e-82 < x

   1. Initial program 12.0%

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

      1. cos-sum 57.1%

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

      2. cancel-sign-sub-inv 57.1%

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

      3. fma-def 57.1%

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

   3. Applied egg-rr 57.1%

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

   4. Taylor expanded in x around inf 57.1%

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

      1. *-commutative 57.1%

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

      2. neg-mul-1 57.1%

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

      3. associate--l+ 98.7%

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

      4. distribute-lft-neg-in 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-def 98.8%

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

      7. *-commutative 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in x around inf 57.1%

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

      1. +-commutative 57.1%

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

      2. *-commutative 57.1%

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

      3. +-commutative 57.1%

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

      4. associate-+r- 98.7%

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

      5. +-commutative 98.7%

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

      6. mul-1-neg 98.7%

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

      7. unsub-neg 98.7%

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

      8. *-rgt-identity 98.7%

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

      9. distribute-lft-out-- 98.7%

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

      10. sub-neg 98.7%

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

      11. metadata-eval 98.7%

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

      12. *-commutative 98.7%

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

   9. Simplified 98.7%

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

   if -2.85000000000000004e-8 < x < 1.79999999999999999e-82

   1. Initial program 72.4%

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

      1. diff-cos 89.7%

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

      2. div-inv 89.7%

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

      3. metadata-eval 89.7%

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

      4. div-inv 89.7%

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

      5. +-commutative 89.7%

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

      6. metadata-eval 89.7%

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

   3. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. +-commutative 89.7%

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

      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. Step-by-step derivation

      1. add-log-exp 71.6%

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

      2. *-commutative 71.6%

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

      3. +-commutative 71.6%

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

      4. +-rgt-identity 71.6%

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

   7. Applied egg-rr 71.6%

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

   8. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. distribute-lft-out 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-8} \lor \neg \left(x \leq 1.8 \cdot 10^{-82}\right):\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) + x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 4: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00165:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.006)
   (- (cos eps) (cos x))
   (if (<= eps 0.00165)
     (- (* -0.5 (* eps (* eps (cos x)))) (* (sin x) eps))
     (* -2.0 (pow (sin (* eps 0.5)) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.006) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 0.00165) {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (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 (eps <= (-0.006d0)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 0.00165d0) then
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (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 (eps <= -0.006) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 0.00165) {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (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 eps <= -0.006:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 0.00165:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (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 (eps <= -0.006)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 0.00165)
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(sin(x) * 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 (eps <= -0.006)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 0.00165)
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (sin(x) * eps);
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.006], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00165], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0060000000000000001

   1. Initial program 62.9%

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

   2. Taylor expanded in x around 0 64.3%

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

   if -0.0060000000000000001 < eps < 0.00165

   1. Initial program 15.1%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. unpow2 99.2%

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

      4. associate-*l* 99.2%

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

   4. Simplified 99.2%

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

   if 0.00165 < eps

   1. Initial program 50.6%

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

      1. diff-cos 50.6%

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

      2. div-inv 50.6%

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

      3. metadata-eval 50.6%

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

      4. div-inv 50.6%

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

      5. +-commutative 50.6%

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

      6. metadata-eval 50.6%

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

   3. Applied egg-rr 50.6%

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

      1. *-commutative 50.6%

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

      2. +-commutative 50.6%

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

      3. associate--l+ 52.6%

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

      4. +-inverses 52.6%

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

      5. distribute-lft-in 52.6%

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

      6. metadata-eval 52.6%

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

      7. *-commutative 52.6%

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

      8. +-commutative 52.6%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in x around 0 53.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00165:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 5: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + (x + eps)))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((eps * 0.5)) * sin((0.5 * (x + (x + eps)))));
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[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

   1. diff-cos 46.2%

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

   2. div-inv 46.2%

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

   3. metadata-eval 46.2%

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

   4. div-inv 46.2%

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

   5. +-commutative 46.2%

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

   6. metadata-eval 46.2%

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

3. Applied egg-rr 46.2%

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

   1. *-commutative 46.2%

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

   2. +-commutative 46.2%

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

   3. associate--l+ 75.0%

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

   4. +-inverses 75.0%

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

   5. distribute-lft-in 75.0%

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

   6. metadata-eval 75.0%

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

   7. *-commutative 75.0%

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

   8. +-commutative 75.0%

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

5. Simplified 75.0%

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

6. Final simplification 75.0%

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

Alternative 6: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-51} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-26}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.1e-51) (not (<= eps 2.5e-26)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-51) || !(eps <= 2.5e-26)) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.1d-51)) .or. (.not. (eps <= 2.5d-26))) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.1e-51) || !(eps <= 2.5e-26)) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.1e-51) or not (eps <= 2.5e-26):
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.1e-51) || !(eps <= 2.5e-26))
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.1e-51) || ~((eps <= 2.5e-26)))
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.1e-51], N[Not[LessEqual[eps, 2.5e-26]], $MachinePrecision]], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.1e-51 or 2.5000000000000001e-26 < eps

   1. Initial program 51.1%

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

      1. diff-cos 57.9%

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

      2. div-inv 57.9%

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

      3. metadata-eval 57.9%

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

      4. div-inv 57.9%

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

      5. +-commutative 57.9%

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

      6. metadata-eval 57.9%

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

   3. Applied egg-rr 57.9%

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

      1. *-commutative 57.9%

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

      2. +-commutative 57.9%

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

      3. associate--l+ 61.1%

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

      4. +-inverses 61.1%

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

      5. distribute-lft-in 61.1%

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

      6. metadata-eval 61.1%

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

      7. *-commutative 61.1%

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

      8. +-commutative 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in x around 0 59.7%

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

   if -1.1e-51 < eps < 2.5000000000000001e-26

   1. Initial program 17.1%

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

   2. Taylor expanded in eps around 0 91.7%

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

      1. associate-*r* 91.7%

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

      2. mul-1-neg 91.7%

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

   4. Simplified 91.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-51} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-26}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 7: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0002:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -1.3 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00122:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos eps) (cos x))))
   (if (<= eps -0.0002)
     t_0
     (if (<= eps -1.3e-49)
       (* eps (* eps -0.5))
       (if (<= eps 0.00122) (* (sin x) (- eps)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) - cos(x);
	double tmp;
	if (eps <= -0.0002) {
		tmp = t_0;
	} else if (eps <= -1.3e-49) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.00122) {
		tmp = sin(x) * -eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(eps) - cos(x)
    if (eps <= (-0.0002d0)) then
        tmp = t_0
    else if (eps <= (-1.3d-49)) then
        tmp = eps * (eps * (-0.5d0))
    else if (eps <= 0.00122d0) then
        tmp = sin(x) * -eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) - Math.cos(x);
	double tmp;
	if (eps <= -0.0002) {
		tmp = t_0;
	} else if (eps <= -1.3e-49) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.00122) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) - math.cos(x)
	tmp = 0
	if eps <= -0.0002:
		tmp = t_0
	elif eps <= -1.3e-49:
		tmp = eps * (eps * -0.5)
	elif eps <= 0.00122:
		tmp = math.sin(x) * -eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) - cos(x))
	tmp = 0.0
	if (eps <= -0.0002)
		tmp = t_0;
	elseif (eps <= -1.3e-49)
		tmp = Float64(eps * Float64(eps * -0.5));
	elseif (eps <= 0.00122)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) - cos(x);
	tmp = 0.0;
	if (eps <= -0.0002)
		tmp = t_0;
	elseif (eps <= -1.3e-49)
		tmp = eps * (eps * -0.5);
	elseif (eps <= 0.00122)
		tmp = sin(x) * -eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0002], t$95$0, If[LessEqual[eps, -1.3e-49], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00122], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.0000000000000001e-4 or 0.00121999999999999995 < eps

   1. Initial program 56.2%

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

   2. Taylor expanded in x around 0 58.2%

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

   if -2.0000000000000001e-4 < eps < -1.29999999999999997e-49

   1. Initial program 3.6%

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

   2. Taylor expanded in x around 0 3.6%

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

   3. Taylor expanded in eps around 0 89.8%

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

      1. *-commutative 89.8%

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

      2. unpow2 89.8%

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

      3. associate-*l* 89.8%

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

   5. Simplified 89.8%

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

   if -1.29999999999999997e-49 < eps < 0.00121999999999999995

   1. Initial program 16.1%

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

   2. Taylor expanded in eps around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. mul-1-neg 87.1%

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

   4. Simplified 87.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0002:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -1.3 \cdot 10^{-49}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00122:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 8: 48.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5.8 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 7.4 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* eps (* eps -0.5))))
   (if (<= eps -0.00018)
     t_0
     (if (<= eps -5.8e-142)
       t_1
       (if (<= eps 7.4e-62) (* x (- eps)) (if (<= eps 9.5e-7) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = eps * (eps * -0.5);
	double tmp;
	if (eps <= -0.00018) {
		tmp = t_0;
	} else if (eps <= -5.8e-142) {
		tmp = t_1;
	} else if (eps <= 7.4e-62) {
		tmp = x * -eps;
	} else if (eps <= 9.5e-7) {
		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 = cos(eps) + (-1.0d0)
    t_1 = eps * (eps * (-0.5d0))
    if (eps <= (-0.00018d0)) then
        tmp = t_0
    else if (eps <= (-5.8d-142)) then
        tmp = t_1
    else if (eps <= 7.4d-62) then
        tmp = x * -eps
    else if (eps <= 9.5d-7) 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 = Math.cos(eps) + -1.0;
	double t_1 = eps * (eps * -0.5);
	double tmp;
	if (eps <= -0.00018) {
		tmp = t_0;
	} else if (eps <= -5.8e-142) {
		tmp = t_1;
	} else if (eps <= 7.4e-62) {
		tmp = x * -eps;
	} else if (eps <= 9.5e-7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	t_1 = eps * (eps * -0.5)
	tmp = 0
	if eps <= -0.00018:
		tmp = t_0
	elif eps <= -5.8e-142:
		tmp = t_1
	elif eps <= 7.4e-62:
		tmp = x * -eps
	elif eps <= 9.5e-7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(eps * Float64(eps * -0.5))
	tmp = 0.0
	if (eps <= -0.00018)
		tmp = t_0;
	elseif (eps <= -5.8e-142)
		tmp = t_1;
	elseif (eps <= 7.4e-62)
		tmp = Float64(x * Float64(-eps));
	elseif (eps <= 9.5e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	t_1 = eps * (eps * -0.5);
	tmp = 0.0;
	if (eps <= -0.00018)
		tmp = t_0;
	elseif (eps <= -5.8e-142)
		tmp = t_1;
	elseif (eps <= 7.4e-62)
		tmp = x * -eps;
	elseif (eps <= 9.5e-7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00018], t$95$0, If[LessEqual[eps, -5.8e-142], t$95$1, If[LessEqual[eps, 7.4e-62], N[(x * (-eps)), $MachinePrecision], If[LessEqual[eps, 9.5e-7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.80000000000000011e-4 or 9.5000000000000001e-7 < eps

   1. Initial program 55.5%

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

   2. Taylor expanded in x around 0 56.8%

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

   if -1.80000000000000011e-4 < eps < -5.7999999999999998e-142 or 7.3999999999999996e-62 < eps < 9.5000000000000001e-7

   1. Initial program 4.1%

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

   2. Taylor expanded in x around 0 4.1%

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

   3. Taylor expanded in eps around 0 48.1%

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

      1. *-commutative 48.1%

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

      2. unpow2 48.1%

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

      3. associate-*l* 48.1%

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

   5. Simplified 48.1%

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

   if -5.7999999999999998e-142 < eps < 7.3999999999999996e-62

   1. Initial program 22.7%

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

      1. add-log-exp 22.0%

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

   3. Applied egg-rr 22.0%

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

   4. Taylor expanded in eps around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. distribute-lft-neg-out 98.3%

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

      3. *-commutative 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in x around 0 39.9%

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

      1. associate-*r* 39.9%

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

      2. neg-mul-1 39.9%

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

   9. Simplified 39.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -5.8 \cdot 10^{-142}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 7.4 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 9: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -2.35 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00125:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)))
   (if (<= eps -0.00018)
     t_0
     (if (<= eps -2.35e-52)
       (* eps (* eps -0.5))
       (if (<= eps 0.00125) (* (sin x) (- eps)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double tmp;
	if (eps <= -0.00018) {
		tmp = t_0;
	} else if (eps <= -2.35e-52) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.00125) {
		tmp = sin(x) * -eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(eps) + (-1.0d0)
    if (eps <= (-0.00018d0)) then
        tmp = t_0
    else if (eps <= (-2.35d-52)) then
        tmp = eps * (eps * (-0.5d0))
    else if (eps <= 0.00125d0) then
        tmp = sin(x) * -eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) + -1.0;
	double tmp;
	if (eps <= -0.00018) {
		tmp = t_0;
	} else if (eps <= -2.35e-52) {
		tmp = eps * (eps * -0.5);
	} else if (eps <= 0.00125) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	tmp = 0
	if eps <= -0.00018:
		tmp = t_0
	elif eps <= -2.35e-52:
		tmp = eps * (eps * -0.5)
	elif eps <= 0.00125:
		tmp = math.sin(x) * -eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	tmp = 0.0
	if (eps <= -0.00018)
		tmp = t_0;
	elseif (eps <= -2.35e-52)
		tmp = Float64(eps * Float64(eps * -0.5));
	elseif (eps <= 0.00125)
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	tmp = 0.0;
	if (eps <= -0.00018)
		tmp = t_0;
	elseif (eps <= -2.35e-52)
		tmp = eps * (eps * -0.5);
	elseif (eps <= 0.00125)
		tmp = sin(x) * -eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[eps, -0.00018], t$95$0, If[LessEqual[eps, -2.35e-52], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00125], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.80000000000000011e-4 or 0.00125000000000000003 < eps

   1. Initial program 56.2%

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

   2. Taylor expanded in x around 0 57.4%

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

   if -1.80000000000000011e-4 < eps < -2.3499999999999999e-52

   1. Initial program 3.6%

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

   2. Taylor expanded in x around 0 3.6%

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

   3. Taylor expanded in eps around 0 89.8%

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

      1. *-commutative 89.8%

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

      2. unpow2 89.8%

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

      3. associate-*l* 89.8%

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

   5. Simplified 89.8%

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

   if -2.3499999999999999e-52 < eps < 0.00125000000000000003

   1. Initial program 16.1%

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

   2. Taylor expanded in eps around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. mul-1-neg 87.1%

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

   4. Simplified 87.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -2.35 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00125:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 10: 24.5% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-65} \lor \neg \left(x \leq 2.3 \cdot 10^{-124}\right):\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -3.4e-65) (not (<= x 2.3e-124)))
   (* x (- eps))
   (* eps (* eps -0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -3.4e-65) || !(x <= 2.3e-124)) {
		tmp = x * -eps;
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-3.4d-65)) .or. (.not. (x <= 2.3d-124))) then
        tmp = x * -eps
    else
        tmp = eps * (eps * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -3.4e-65) || !(x <= 2.3e-124)) {
		tmp = x * -eps;
	} else {
		tmp = eps * (eps * -0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -3.4e-65) or not (x <= 2.3e-124):
		tmp = x * -eps
	else:
		tmp = eps * (eps * -0.5)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -3.4e-65) || !(x <= 2.3e-124))
		tmp = Float64(x * Float64(-eps));
	else
		tmp = Float64(eps * Float64(eps * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -3.4e-65) || ~((x <= 2.3e-124)))
		tmp = x * -eps;
	else
		tmp = eps * (eps * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -3.4e-65], N[Not[LessEqual[x, 2.3e-124]], $MachinePrecision]], N[(x * (-eps)), $MachinePrecision], N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999987e-65 or 2.30000000000000012e-124 < x

   1. Initial program 19.9%

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

      1. add-log-exp 19.9%

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

   3. Applied egg-rr 19.9%

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

   4. Taylor expanded in eps around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. distribute-lft-neg-out 45.6%

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

      3. *-commutative 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in x around 0 12.1%

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

      1. associate-*r* 12.1%

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

      2. neg-mul-1 12.1%

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

   9. Simplified 12.1%

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

   if -3.39999999999999987e-65 < x < 2.30000000000000012e-124

   1. Initial program 75.9%

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

   2. Taylor expanded in x around 0 75.9%

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

   3. Taylor expanded in eps around 0 37.1%

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

      1. *-commutative 37.1%

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

      2. unpow2 37.1%

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

      3. associate-*l* 37.1%

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

   5. Simplified 37.1%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-65} \lor \neg \left(x \leq 2.3 \cdot 10^{-124}\right):\\ \;\;\;\;x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 11: 18.9% 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 38.9%

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

   1. add-log-exp 38.8%

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

3. Applied egg-rr 38.8%

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

4. Taylor expanded in eps around 0 36.5%

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

   1. mul-1-neg 36.5%

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

   2. distribute-lft-neg-out 36.5%

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

   3. *-commutative 36.5%

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

6. Simplified 36.5%

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

7. Taylor expanded in x around 0 14.3%

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

   1. associate-*r* 14.3%

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

   2. neg-mul-1 14.3%

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

9. Simplified 14.3%

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

10. Final simplification 14.3%

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

Reproduce

herbie shell --seed 2023229 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))