2cos (problem 3.3.5)

Percentage Accurate: 52.7% → 99.7%

Time: 19.4s

Alternatives: 8

Speedup: 51.3×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

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: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (sin x)
  (- (sin eps))
  (*
   (cos x)
   (+ (* -0.5 (pow eps 2.0)) (* 0.041666666666666664 (pow eps 4.0))))))

C

double code(double x, double eps) {
	return fma(sin(x), -sin(eps), (cos(x) * ((-0.5 * pow(eps, 2.0)) + (0.041666666666666664 * pow(eps, 4.0)))));
}

Julia

function code(x, eps)
	return fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(Float64(-0.5 * (eps ^ 2.0)) + Float64(0.041666666666666664 * (eps ^ 4.0)))))
end

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. cos-sum 55.2%

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

   2. cancel-sign-sub-inv 55.2%

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

   3. add-cube-cbrt 55.2%

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

   4. associate-*l* 55.2%

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

   5. fma-def 55.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x} \cdot \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   6. pow2 55.2%

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

4. Applied egg-rr 55.2%

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

5. Taylor expanded in x around inf 55.2%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin \varepsilon \cdot \sin x\right) + {1}^{0.3333333333333333} \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) - \cos x} \]
6. Step-by-step derivation

   1. associate--l+ 81.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\sin \varepsilon \cdot \sin x\right) + \left({1}^{0.3333333333333333} \cdot \left(\cos \varepsilon \cdot \cos x\right) - \cos x\right)} \]

   2. *-commutative 81.5%

      \[\leadsto -1 \cdot \color{blue}{\left(\sin x \cdot \sin \varepsilon\right)} + \left({1}^{0.3333333333333333} \cdot \left(\cos \varepsilon \cdot \cos x\right) - \cos x\right) \]

   3. neg-mul-1 81.5%

      \[\leadsto \color{blue}{\left(-\sin x \cdot \sin \varepsilon\right)} + \left({1}^{0.3333333333333333} \cdot \left(\cos \varepsilon \cdot \cos x\right) - \cos x\right) \]

   4. distribute-rgt-neg-in 81.5%

      \[\leadsto \color{blue}{\sin x \cdot \left(-\sin \varepsilon\right)} + \left({1}^{0.3333333333333333} \cdot \left(\cos \varepsilon \cdot \cos x\right) - \cos x\right) \]

   5. fma-def 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, -\sin \varepsilon, {1}^{0.3333333333333333} \cdot \left(\cos \varepsilon \cdot \cos x\right) - \cos x\right)} \]

   6. pow-base-1 81.5%

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

   7. associate-*r* 81.5%

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

   8. *-lft-identity 81.5%

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

   9. *-commutative 81.5%

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

   10. *-rgt-identity 81.5%

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

   11. distribute-lft-out-- 81.5%

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

   12. sub-neg 81.5%

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

   13. metadata-eval 81.5%

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

   14. +-commutative 81.5%

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

7. Simplified 81.5%

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

8. Taylor expanded in eps around 0 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. diff-cos 81.3%

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

   2. div-inv 81.3%

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

   3. associate--l+ 81.3%

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

   4. metadata-eval 81.3%

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

   5. div-inv 81.3%

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

   6. +-commutative 81.3%

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

   7. associate-+l+ 81.3%

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

   8. metadata-eval 81.3%

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

4. Applied egg-rr 81.3%

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

   1. associate-*r* 81.4%

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

   2. *-commutative 81.4%

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

   3. *-commutative 81.4%

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

   4. +-commutative 81.4%

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

   5. count-2 81.4%

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

   6. fma-def 81.4%

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

   7. sub-neg 81.4%

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

   8. mul-1-neg 81.4%

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

   9. +-commutative 81.4%

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

   10. associate-+r+ 99.7%

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

   11. mul-1-neg 99.7%

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

   12. sub-neg 99.7%

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

   13. +-inverses 99.7%

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

   14. remove-double-neg 99.7%

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

   15. mul-1-neg 99.7%

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

   16. sub-neg 99.7%

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

   17. neg-sub0 99.7%

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

   18. mul-1-neg 99.7%

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

   19. remove-double-neg 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(-2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.1%

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

   1. diff-cos 81.3%

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

   2. *-commutative 81.3%

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

   3. div-inv 81.3%

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

   4. associate--l+ 81.3%

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

   5. metadata-eval 81.3%

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

   6. div-inv 81.3%

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

   7. +-commutative 81.3%

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

   8. associate-+l+ 81.3%

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

   9. metadata-eval 81.3%

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

4. Applied egg-rr 81.3%

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

5. Taylor expanded in x around 0 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. mul-1-neg 99.3%

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

   3. unsub-neg 99.3%

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

   4. associate-*r* 99.3%

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

   5. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 97.3%

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

7. Final simplification 97.3%

   \[\leadsto \cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2}\right) - x \cdot \varepsilon \]
8. Add Preprocessing

Alternative 5: 97.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot {\varepsilon}^{2} - x \cdot \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. mul-1-neg 99.3%

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

   3. unsub-neg 99.3%

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

   4. associate-*r* 99.3%

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

   5. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 97.3%

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

7. Taylor expanded in x around 0 97.3%

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

   1. mul-1-neg 97.3%

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

   2. distribute-lft-neg-out 97.3%

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

   3. +-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. *-commutative 97.3%

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

   6. distribute-rgt-neg-in 97.3%

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

   7. unsub-neg 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

    \[\leadsto -0.5 \cdot {\varepsilon}^{2} - x \cdot \varepsilon \]
11. Add Preprocessing

Alternative 6: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 81.0%

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

   1. mul-1-neg 81.0%

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

   2. *-commutative 81.0%

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

   3. distribute-rgt-neg-in 81.0%

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

5. Simplified 81.0%

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

6. Final simplification 81.0%

   \[\leadsto \sin x \cdot \left(-\varepsilon\right) \]
7. Add Preprocessing

Alternative 7: 78.5% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 81.0%

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

   1. mul-1-neg 81.0%

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

   2. *-commutative 81.0%

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

   3. distribute-rgt-neg-in 81.0%

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

5. Simplified 81.0%

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

6. Taylor expanded in x around 0 79.9%

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

   1. *-commutative 79.9%

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

   2. associate-*r* 79.9%

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

   3. neg-mul-1 79.9%

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

8. Simplified 79.9%

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

9. Final simplification 79.9%

   \[\leadsto x \cdot \left(-\varepsilon\right) \]
10. Add Preprocessing

Alternative 8: 51.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 55.1%

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

   1. cos-sum 55.2%

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

   2. cancel-sign-sub-inv 55.2%

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

   3. add-cube-cbrt 55.2%

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

   4. associate-*l* 55.2%

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

   5. fma-def 55.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}, \sqrt[3]{\cos x} \cdot \cos \varepsilon, \left(-\sin x\right) \cdot \sin \varepsilon\right)} - \cos x \]

   6. pow2 55.2%

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

4. Applied egg-rr 55.2%

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

5. Taylor expanded in eps around 0 53.7%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \cos x - \cos x} \]
6. Step-by-step derivation

   1. pow-base-1 53.7%

      \[\leadsto \color{blue}{1} \cdot \cos x - \cos x \]

   2. *-lft-identity 53.7%

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

   3. +-inverses 53.7%

      \[\leadsto \color{blue}{0} \]

7. Simplified 53.7%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 53.7%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :herbie-target
  (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

  (- (cos (+ x eps)) (cos x)))