2cos (problem 3.3.5)

Percentage Accurate: 38.6% → 98.8%

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: 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -7.8)
   (- (fma (cos x) (cos eps) (* (- (sin x)) (sin eps))) (cos x))
   (if (<= eps 0.00019)
     (* (sin (/ (+ eps (- x x)) 2.0)) (* (sin (/ (+ eps (+ x x)) 2.0)) -2.0))
     (fma (cos x) (cos eps) (- (fma (sin x) (sin eps) (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -7.8) {
		tmp = fma(cos(x), cos(eps), (-sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.00019) {
		tmp = sin(((eps + (x - x)) / 2.0)) * (sin(((eps + (x + x)) / 2.0)) * -2.0);
	} else {
		tmp = fma(cos(x), cos(eps), -fma(sin(x), sin(eps), cos(x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -7.8)
		tmp = Float64(fma(cos(x), cos(eps), Float64(Float64(-sin(x)) * sin(eps))) - cos(x));
	elseif (eps <= 0.00019)
		tmp = Float64(sin(Float64(Float64(eps + Float64(x - x)) / 2.0)) * Float64(sin(Float64(Float64(eps + Float64(x + x)) / 2.0)) * -2.0));
	else
		tmp = fma(cos(x), cos(eps), Float64(-fma(sin(x), sin(eps), cos(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -7.8], 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], If[LessEqual[eps, 0.00019], N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision] + (-N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.79999999999999982

   1. Initial program 54.3%

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

      1. cos-sum 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. fma-def 98.7%

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

   3. Applied egg-rr 98.7%

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

   if -7.79999999999999982 < eps < 1.9000000000000001e-4

   1. Initial program 35.6%

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

      1. add-exp-log 34.0%

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

   3. Applied egg-rr 34.0%

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

      1. add-exp-log 35.6%

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

      2. diff-cos 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 52.2%

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

      3. +-commutative 52.2%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   7. Simplified 99.7%

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

   if 1.9000000000000001e-4 < eps

   1. Initial program 48.0%

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

      1. cos-sum 98.6%

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

      2. associate--l- 98.7%

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

      3. fma-neg 98.8%

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

      4. fma-def 98.9%

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

   3. Applied egg-rr 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00019:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\left(t_0 - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))))
   (if (<= eps -7.8)
     (- (- t_0 (* (sin x) (sin eps))) (cos x))
     (if (<= eps 0.0054)
       (* (sin (/ (+ eps (- x x)) 2.0)) (* (sin (/ (+ eps (+ x x)) 2.0)) -2.0))
       (- t_0 (fma (sin eps) (sin x) (cos x)))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double tmp;
	if (eps <= -7.8) {
		tmp = (t_0 - (sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.0054) {
		tmp = sin(((eps + (x - x)) / 2.0)) * (sin(((eps + (x + x)) / 2.0)) * -2.0);
	} else {
		tmp = t_0 - fma(sin(eps), sin(x), cos(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * cos(eps))
	tmp = 0.0
	if (eps <= -7.8)
		tmp = Float64(Float64(t_0 - Float64(sin(x) * sin(eps))) - cos(x));
	elseif (eps <= 0.0054)
		tmp = Float64(sin(Float64(Float64(eps + Float64(x - x)) / 2.0)) * Float64(sin(Float64(Float64(eps + Float64(x + x)) / 2.0)) * -2.0));
	else
		tmp = Float64(t_0 - fma(sin(eps), sin(x), cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -7.8], N[(N[(t$95$0 - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0054], N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.79999999999999982

   1. Initial program 54.3%

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

      1. cos-sum 98.7%

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

   3. Applied egg-rr 98.7%

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

   if -7.79999999999999982 < eps < 0.0054000000000000003

   1. Initial program 35.6%

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

      1. add-exp-log 34.0%

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

   3. Applied egg-rr 34.0%

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

      1. add-exp-log 35.6%

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

      2. diff-cos 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 52.2%

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

      3. +-commutative 52.2%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   7. Simplified 99.7%

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

   if 0.0054000000000000003 < eps

   1. Initial program 48.0%

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

      1. sub-neg 48.0%

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

      2. cos-sum 98.6%

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

      3. associate-+l- 98.7%

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

      4. fma-neg 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. fma-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternative 3: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -7.8)
   (- (fma (cos x) (cos eps) (* (- (sin x)) (sin eps))) (cos x))
   (if (<= eps 0.0054)
     (* (sin (/ (+ eps (- x x)) 2.0)) (* (sin (/ (+ eps (+ x x)) 2.0)) -2.0))
     (- (* (cos x) (cos eps)) (fma (sin eps) (sin x) (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -7.8) {
		tmp = fma(cos(x), cos(eps), (-sin(x) * sin(eps))) - cos(x);
	} else if (eps <= 0.0054) {
		tmp = sin(((eps + (x - x)) / 2.0)) * (sin(((eps + (x + x)) / 2.0)) * -2.0);
	} else {
		tmp = (cos(x) * cos(eps)) - fma(sin(eps), sin(x), cos(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -7.8)
		tmp = Float64(fma(cos(x), cos(eps), Float64(Float64(-sin(x)) * sin(eps))) - cos(x));
	elseif (eps <= 0.0054)
		tmp = Float64(sin(Float64(Float64(eps + Float64(x - x)) / 2.0)) * Float64(sin(Float64(Float64(eps + Float64(x + x)) / 2.0)) * -2.0));
	else
		tmp = Float64(Float64(cos(x) * cos(eps)) - fma(sin(eps), sin(x), cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -7.8], 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], If[LessEqual[eps, 0.0054], N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.79999999999999982

   1. Initial program 54.3%

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

      1. cos-sum 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. fma-def 98.7%

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

   3. Applied egg-rr 98.7%

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

   if -7.79999999999999982 < eps < 0.0054000000000000003

   1. Initial program 35.6%

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

      1. add-exp-log 34.0%

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

   3. Applied egg-rr 34.0%

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

      1. add-exp-log 35.6%

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

      2. diff-cos 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 52.2%

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

      3. +-commutative 52.2%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   7. Simplified 99.7%

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

   if 0.0054000000000000003 < eps

   1. Initial program 48.0%

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

      1. sub-neg 48.0%

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

      2. cos-sum 98.6%

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

      3. associate-+l- 98.7%

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

      4. fma-neg 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. fma-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-neg 98.8%

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

      5. remove-double-neg 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8 \lor \neg \left(\varepsilon \leq 0.00096\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -7.8) (not (<= eps 0.00096)))
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
   (* (sin (/ (+ eps (- x x)) 2.0)) (* (sin (/ (+ eps (+ x x)) 2.0)) -2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.8) || !(eps <= 0.00096)) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else {
		tmp = sin(((eps + (x - x)) / 2.0)) * (sin(((eps + (x + x)) / 2.0)) * -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 <= (-7.8d0)) .or. (.not. (eps <= 0.00096d0))) then
        tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)))
    else
        tmp = sin(((eps + (x - x)) / 2.0d0)) * (sin(((eps + (x + x)) / 2.0d0)) * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.8) || !(eps <= 0.00096)) {
		tmp = (Math.cos(x) * Math.cos(eps)) - (Math.cos(x) + (Math.sin(x) * Math.sin(eps)));
	} else {
		tmp = Math.sin(((eps + (x - x)) / 2.0)) * (Math.sin(((eps + (x + x)) / 2.0)) * -2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -7.8) or not (eps <= 0.00096):
		tmp = (math.cos(x) * math.cos(eps)) - (math.cos(x) + (math.sin(x) * math.sin(eps)))
	else:
		tmp = math.sin(((eps + (x - x)) / 2.0)) * (math.sin(((eps + (x + x)) / 2.0)) * -2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -7.8) || !(eps <= 0.00096))
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	else
		tmp = Float64(sin(Float64(Float64(eps + Float64(x - x)) / 2.0)) * Float64(sin(Float64(Float64(eps + Float64(x + x)) / 2.0)) * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -7.8) || ~((eps <= 0.00096)))
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	else
		tmp = sin(((eps + (x - x)) / 2.0)) * (sin(((eps + (x + x)) / 2.0)) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -7.8], N[Not[LessEqual[eps, 0.00096]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.79999999999999982 or 9.60000000000000024e-4 < eps

   1. Initial program 51.2%

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

      1. cos-sum 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. fma-def 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. +-commutative 98.7%

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

      3. sub-neg 98.7%

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

      4. associate--l- 98.6%

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

      5. *-commutative 98.6%

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

      6. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   if -7.79999999999999982 < eps < 9.60000000000000024e-4

   1. Initial program 35.6%

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

      1. add-exp-log 34.0%

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

   3. Applied egg-rr 34.0%

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

      1. add-exp-log 35.6%

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

      2. diff-cos 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 52.2%

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

      3. +-commutative 52.2%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8 \lor \neg \left(\varepsilon \leq 0.00096\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00095:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin x) (sin eps))))
   (if (<= eps -7.8)
     (- (- t_0 t_1) (cos x))
     (if (<= eps 0.00095)
       (* (sin (/ (+ eps (- x x)) 2.0)) (* (sin (/ (+ eps (+ x x)) 2.0)) -2.0))
       (- t_0 (+ (cos x) t_1))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double t_1 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -7.8) {
		tmp = (t_0 - t_1) - cos(x);
	} else if (eps <= 0.00095) {
		tmp = sin(((eps + (x - x)) / 2.0)) * (sin(((eps + (x + x)) / 2.0)) * -2.0);
	} else {
		tmp = t_0 - (cos(x) + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(x) * cos(eps)
    t_1 = sin(x) * sin(eps)
    if (eps <= (-7.8d0)) then
        tmp = (t_0 - t_1) - cos(x)
    else if (eps <= 0.00095d0) then
        tmp = sin(((eps + (x - x)) / 2.0d0)) * (sin(((eps + (x + x)) / 2.0d0)) * (-2.0d0))
    else
        tmp = t_0 - (cos(x) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(x) * Math.cos(eps);
	double t_1 = Math.sin(x) * Math.sin(eps);
	double tmp;
	if (eps <= -7.8) {
		tmp = (t_0 - t_1) - Math.cos(x);
	} else if (eps <= 0.00095) {
		tmp = Math.sin(((eps + (x - x)) / 2.0)) * (Math.sin(((eps + (x + x)) / 2.0)) * -2.0);
	} else {
		tmp = t_0 - (Math.cos(x) + t_1);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(x) * math.cos(eps)
	t_1 = math.sin(x) * math.sin(eps)
	tmp = 0
	if eps <= -7.8:
		tmp = (t_0 - t_1) - math.cos(x)
	elif eps <= 0.00095:
		tmp = math.sin(((eps + (x - x)) / 2.0)) * (math.sin(((eps + (x + x)) / 2.0)) * -2.0)
	else:
		tmp = t_0 - (math.cos(x) + t_1)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * cos(eps))
	t_1 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -7.8)
		tmp = Float64(Float64(t_0 - t_1) - cos(x));
	elseif (eps <= 0.00095)
		tmp = Float64(sin(Float64(Float64(eps + Float64(x - x)) / 2.0)) * Float64(sin(Float64(Float64(eps + Float64(x + x)) / 2.0)) * -2.0));
	else
		tmp = Float64(t_0 - Float64(cos(x) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(x) * cos(eps);
	t_1 = sin(x) * sin(eps);
	tmp = 0.0;
	if (eps <= -7.8)
		tmp = (t_0 - t_1) - cos(x);
	elseif (eps <= 0.00095)
		tmp = sin(((eps + (x - x)) / 2.0)) * (sin(((eps + (x + x)) / 2.0)) * -2.0);
	else
		tmp = t_0 - (cos(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -7.8], N[(N[(t$95$0 - t$95$1), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00095], N[(N[Sin[N[(N[(eps + N[(x - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.79999999999999982

   1. Initial program 54.3%

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

      1. cos-sum 98.7%

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

   3. Applied egg-rr 98.7%

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

   if -7.79999999999999982 < eps < 9.49999999999999998e-4

   1. Initial program 35.6%

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

      1. add-exp-log 34.0%

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

   3. Applied egg-rr 34.0%

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

      1. add-exp-log 35.6%

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

      2. diff-cos 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 52.2%

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

      3. +-commutative 52.2%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

   7. Simplified 99.7%

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

   if 9.49999999999999998e-4 < eps

   1. Initial program 48.0%

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

      1. cos-sum 98.6%

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

      2. cancel-sign-sub-inv 98.6%

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

      3. fma-def 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in x around inf 98.6%

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

      1. neg-mul-1 98.6%

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

      2. +-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. associate--l- 98.7%

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

      5. *-commutative 98.7%

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

      6. *-commutative 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.8:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00095:\\ \;\;\;\;\sin \left(\frac{\varepsilon + \left(x - x\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon + \left(x + x\right)}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 6: 76.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.1%

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

   1. add-exp-log 29.9%

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

3. Applied egg-rr 29.9%

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

   1. add-exp-log 43.1%

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

   2. diff-cos 51.5%

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

5. Applied egg-rr 51.5%

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

   1. *-commutative 51.5%

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

   2. associate-*l* 51.5%

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

   3. +-commutative 51.5%

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

   4. associate--l+ 76.6%

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

   5. +-commutative 76.6%

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

   6. associate-+l+ 76.6%

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

7. Simplified 76.6%

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

8. Final simplification 76.6%

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

Alternative 7: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (eps - (x * (-2.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.1%

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

   1. add-exp-log 29.9%

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

3. Applied egg-rr 29.9%

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

   1. add-exp-log 43.1%

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

   2. diff-cos 51.5%

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

5. Applied egg-rr 51.5%

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

   1. *-commutative 51.5%

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

   2. associate-*l* 51.5%

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

   3. +-commutative 51.5%

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

   4. associate--l+ 76.6%

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

   5. +-commutative 76.6%

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

   6. associate-+l+ 76.6%

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

7. Simplified 76.6%

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

8. Taylor expanded in x around -inf 76.6%

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

9. Final simplification 76.6%

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

Alternative 8: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ t_1 := -0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.4 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-79}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.7:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos eps) (cos x))) (t_1 (* -0.5 (* eps (+ x (+ eps x))))))
   (if (<= eps -5e-5)
     t_0
     (if (<= eps -3.4e-71)
       t_1
       (if (<= eps 3.5e-79) (* eps (- (sin x))) (if (<= eps 4.7) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) - cos(x);
	double t_1 = -0.5 * (eps * (x + (eps + x)));
	double tmp;
	if (eps <= -5e-5) {
		tmp = t_0;
	} else if (eps <= -3.4e-71) {
		tmp = t_1;
	} else if (eps <= 3.5e-79) {
		tmp = eps * -sin(x);
	} else if (eps <= 4.7) {
		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) - cos(x)
    t_1 = (-0.5d0) * (eps * (x + (eps + x)))
    if (eps <= (-5d-5)) then
        tmp = t_0
    else if (eps <= (-3.4d-71)) then
        tmp = t_1
    else if (eps <= 3.5d-79) then
        tmp = eps * -sin(x)
    else if (eps <= 4.7d0) 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) - Math.cos(x);
	double t_1 = -0.5 * (eps * (x + (eps + x)));
	double tmp;
	if (eps <= -5e-5) {
		tmp = t_0;
	} else if (eps <= -3.4e-71) {
		tmp = t_1;
	} else if (eps <= 3.5e-79) {
		tmp = eps * -Math.sin(x);
	} else if (eps <= 4.7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) - math.cos(x)
	t_1 = -0.5 * (eps * (x + (eps + x)))
	tmp = 0
	if eps <= -5e-5:
		tmp = t_0
	elif eps <= -3.4e-71:
		tmp = t_1
	elif eps <= 3.5e-79:
		tmp = eps * -math.sin(x)
	elif eps <= 4.7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) - cos(x))
	t_1 = Float64(-0.5 * Float64(eps * Float64(x + Float64(eps + x))))
	tmp = 0.0
	if (eps <= -5e-5)
		tmp = t_0;
	elseif (eps <= -3.4e-71)
		tmp = t_1;
	elseif (eps <= 3.5e-79)
		tmp = Float64(eps * Float64(-sin(x)));
	elseif (eps <= 4.7)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) - cos(x);
	t_1 = -0.5 * (eps * (x + (eps + x)));
	tmp = 0.0;
	if (eps <= -5e-5)
		tmp = t_0;
	elseif (eps <= -3.4e-71)
		tmp = t_1;
	elseif (eps <= 3.5e-79)
		tmp = eps * -sin(x);
	elseif (eps <= 4.7)
		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] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(eps * N[(x + N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5e-5], t$95$0, If[LessEqual[eps, -3.4e-71], t$95$1, If[LessEqual[eps, 3.5e-79], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 4.7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.00000000000000024e-5 or 4.70000000000000018 < eps

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 53.8%

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

   if -5.00000000000000024e-5 < eps < -3.40000000000000003e-71 or 3.5000000000000003e-79 < eps < 4.70000000000000018

   1. Initial program 7.6%

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

   2. Taylor expanded in x around 0 7.6%

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

   3. Taylor expanded in eps around 0 65.3%

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

      1. +-rgt-identity 65.3%

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

      2. metadata-eval 65.3%

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

      3. +-inverses 65.3%

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

      4. +-inverses 65.3%

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

      5. flip-- 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. metadata-eval 0.0%

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

      12. metadata-eval 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. metadata-eval 0.0%

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

      17. +-inverses 0.0%

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

      18. flip-+ 64.6%

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

      19. pow2 64.6%

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

      20. associate-+r+ 64.6%

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

      21. +-commutative 64.6%

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

      22. distribute-rgt-in 64.6%

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

   5. Applied egg-rr 72.5%

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

      1. distribute-rgt-out 72.5%

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

      2. +-commutative 72.5%

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

   7. Simplified 72.5%

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

   if -3.40000000000000003e-71 < eps < 3.5000000000000003e-79

   1. Initial program 42.2%

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

   2. Taylor expanded in eps around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq -3.4 \cdot 10^{-71}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-79}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.7:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 9: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-69} \lor \neg \left(\varepsilon \leq 8 \cdot 10^{-30}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3e-69) (not (<= eps 8e-30)))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-69) || !(eps <= 8e-30)) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = eps * -sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3d-69)) .or. (.not. (eps <= 8d-30))) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-69) || !(eps <= 8e-30)) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3e-69) or not (eps <= 8e-30):
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3e-69) || !(eps <= 8e-30))
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3e-69) || ~((eps <= 8e-30)))
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3e-69], N[Not[LessEqual[eps, 8e-30]], $MachinePrecision]], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.99999999999999989e-69 or 8.000000000000001e-30 < eps

   1. Initial program 46.4%

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

      1. diff-cos 55.8%

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

      2. div-inv 55.8%

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

      3. associate--l+ 55.8%

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

      4. metadata-eval 55.8%

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

      5. div-inv 55.8%

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

      6. +-commutative 55.8%

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

      7. associate-+l+ 55.9%

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

      8. metadata-eval 55.9%

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

   3. Applied egg-rr 55.9%

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

      1. associate-*r* 55.9%

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

      2. *-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

      5. count-2 55.9%

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

      6. fma-def 55.9%

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

      7. sub-neg 55.9%

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

      8. mul-1-neg 55.9%

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

      9. +-commutative 55.9%

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

      10. associate-+r+ 58.7%

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

      11. mul-1-neg 58.7%

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

      12. sub-neg 58.7%

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

      13. +-inverses 58.7%

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

      14. remove-double-neg 58.7%

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

      15. mul-1-neg 58.7%

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

      16. sub-neg 58.7%

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

      17. neg-sub0 58.7%

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

      18. mul-1-neg 58.7%

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

      19. remove-double-neg 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in x around 0 55.7%

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

   if -2.99999999999999989e-69 < eps < 8.000000000000001e-30

   1. Initial program 38.8%

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

   2. Taylor expanded in eps around 0 93.1%

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

      1. mul-1-neg 93.1%

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

      2. *-commutative 93.1%

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

      3. distribute-rgt-neg-in 93.1%

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

   4. Simplified 93.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-69} \lor \neg \left(\varepsilon \leq 8 \cdot 10^{-30}\right):\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := -0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \mathbf{if}\;\varepsilon \leq -0.006:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.2 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-79}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.7:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* -0.5 (* eps (+ x (+ eps x))))))
   (if (<= eps -0.006)
     t_0
     (if (<= eps -3.2e-72)
       t_1
       (if (<= eps 1.6e-79) (* eps (- (sin x))) (if (<= eps 4.7) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = -0.5 * (eps * (x + (eps + x)));
	double tmp;
	if (eps <= -0.006) {
		tmp = t_0;
	} else if (eps <= -3.2e-72) {
		tmp = t_1;
	} else if (eps <= 1.6e-79) {
		tmp = eps * -sin(x);
	} else if (eps <= 4.7) {
		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 = (-0.5d0) * (eps * (x + (eps + x)))
    if (eps <= (-0.006d0)) then
        tmp = t_0
    else if (eps <= (-3.2d-72)) then
        tmp = t_1
    else if (eps <= 1.6d-79) then
        tmp = eps * -sin(x)
    else if (eps <= 4.7d0) 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 = -0.5 * (eps * (x + (eps + x)));
	double tmp;
	if (eps <= -0.006) {
		tmp = t_0;
	} else if (eps <= -3.2e-72) {
		tmp = t_1;
	} else if (eps <= 1.6e-79) {
		tmp = eps * -Math.sin(x);
	} else if (eps <= 4.7) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	t_1 = -0.5 * (eps * (x + (eps + x)))
	tmp = 0
	if eps <= -0.006:
		tmp = t_0
	elif eps <= -3.2e-72:
		tmp = t_1
	elif eps <= 1.6e-79:
		tmp = eps * -math.sin(x)
	elif eps <= 4.7:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(-0.5 * Float64(eps * Float64(x + Float64(eps + x))))
	tmp = 0.0
	if (eps <= -0.006)
		tmp = t_0;
	elseif (eps <= -3.2e-72)
		tmp = t_1;
	elseif (eps <= 1.6e-79)
		tmp = Float64(eps * Float64(-sin(x)));
	elseif (eps <= 4.7)
		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 = -0.5 * (eps * (x + (eps + x)));
	tmp = 0.0;
	if (eps <= -0.006)
		tmp = t_0;
	elseif (eps <= -3.2e-72)
		tmp = t_1;
	elseif (eps <= 1.6e-79)
		tmp = eps * -sin(x);
	elseif (eps <= 4.7)
		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[(-0.5 * N[(eps * N[(x + N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.006], t$95$0, If[LessEqual[eps, -3.2e-72], t$95$1, If[LessEqual[eps, 1.6e-79], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 4.7], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0060000000000000001 or 4.70000000000000018 < eps

   1. Initial program 52.2%

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

   2. Taylor expanded in x around 0 52.6%

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

   if -0.0060000000000000001 < eps < -3.19999999999999999e-72 or 1.59999999999999994e-79 < eps < 4.70000000000000018

   1. Initial program 9.5%

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

   2. Taylor expanded in x around 0 7.4%

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

   3. Taylor expanded in eps around 0 63.2%

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

      1. +-rgt-identity 63.2%

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

      2. metadata-eval 63.2%

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

      3. +-inverses 63.2%

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

      4. +-inverses 63.2%

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

      5. flip-- 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. metadata-eval 0.0%

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

      12. metadata-eval 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. metadata-eval 0.0%

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

      17. +-inverses 0.0%

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

      18. flip-+ 62.6%

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

      19. pow2 62.6%

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

      20. associate-+r+ 62.6%

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

      21. +-commutative 62.6%

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

      22. distribute-rgt-in 62.6%

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

   5. Applied egg-rr 70.6%

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

      1. distribute-rgt-out 70.6%

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

      2. +-commutative 70.6%

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

   7. Simplified 70.6%

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

   if -3.19999999999999999e-72 < eps < 1.59999999999999994e-79

   1. Initial program 42.2%

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

   2. Taylor expanded in eps around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -3.2 \cdot 10^{-72}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-79}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.7:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 11: 52.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006 \lor \neg \left(\varepsilon \leq 4.7\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.006) (not (<= eps 4.7)))
   (+ (cos eps) -1.0)
   (* -0.5 (* eps (+ x (+ eps x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.006) || !(eps <= 4.7)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = -0.5 * (eps * (x + (eps + 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 <= (-0.006d0)) .or. (.not. (eps <= 4.7d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = (-0.5d0) * (eps * (x + (eps + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.006) || !(eps <= 4.7)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = -0.5 * (eps * (x + (eps + x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.006) or not (eps <= 4.7):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = -0.5 * (eps * (x + (eps + x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.006) || !(eps <= 4.7))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(-0.5 * Float64(eps * Float64(x + Float64(eps + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.006) || ~((eps <= 4.7)))
		tmp = cos(eps) + -1.0;
	else
		tmp = -0.5 * (eps * (x + (eps + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.006], N[Not[LessEqual[eps, 4.7]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.5 * N[(eps * N[(x + N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0060000000000000001 or 4.70000000000000018 < eps

   1. Initial program 52.2%

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

   2. Taylor expanded in x around 0 52.6%

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

   if -0.0060000000000000001 < eps < 4.70000000000000018

   1. Initial program 34.6%

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

   2. Taylor expanded in x around 0 34.0%

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

   3. Taylor expanded in eps around 0 49.0%

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

      1. +-rgt-identity 49.0%

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

      2. metadata-eval 49.0%

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

      3. +-inverses 49.0%

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

      4. +-inverses 49.0%

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

      5. flip-- 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. metadata-eval 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. metadata-eval 0.0%

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

      12. metadata-eval 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. metadata-eval 0.0%

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

      17. +-inverses 0.0%

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

      18. flip-+ 35.3%

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

      19. pow2 35.3%

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

      20. associate-+r+ 35.3%

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

      21. +-commutative 35.3%

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

      22. distribute-rgt-in 35.3%

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

   5. Applied egg-rr 61.9%

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

      1. distribute-rgt-out 61.9%

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

      2. +-commutative 61.9%

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

   7. Simplified 61.9%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006 \lor \neg \left(\varepsilon \leq 4.7\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right)\right)\\ \end{array} \]

Alternative 12: 27.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -0.5 * (eps * (x + (eps + x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.1%

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

2. Taylor expanded in x around 0 43.0%

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

3. Taylor expanded in eps around 0 27.0%

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

   1. +-rgt-identity 27.0%

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

   2. metadata-eval 27.0%

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

   3. +-inverses 27.0%

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

   4. +-inverses 27.0%

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

   5. flip-- 0.0%

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

   6. +-inverses 0.0%

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

   7. +-inverses 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. metadata-eval 0.0%

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

   12. metadata-eval 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. +-inverses 0.0%

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

   16. metadata-eval 0.0%

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

   17. +-inverses 0.0%

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

   18. flip-+ 19.8%

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

   19. pow2 19.8%

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

   20. associate-+r+ 19.8%

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

   21. +-commutative 19.8%

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

   22. distribute-rgt-in 19.8%

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

5. Applied egg-rr 33.5%

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

   1. distribute-rgt-out 33.6%

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

   2. +-commutative 33.6%

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

7. Simplified 33.6%

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

8. Final simplification 33.6%

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

Alternative 13: 21.9% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -0.5 * (eps * eps)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.1%

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

2. Taylor expanded in x around 0 43.0%

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

3. Taylor expanded in eps around 0 27.0%

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

   1. unpow2 27.0%

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

5. Applied egg-rr 27.0%

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

6. Final simplification 27.0%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))