2cos (problem 3.3.5)

Percentage Accurate: 37.8% → 99.5%

Time: 16.6s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.8% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (sin x)
  (- (sin eps))
  (* (cos x) (* (/ (sin eps) -1.0) (tan (/ eps 2.0))))))

C

double code(double x, double eps) {
	return fma(sin(x), -sin(eps), (cos(x) * ((sin(eps) / -1.0) * tan((eps / 2.0)))));
}

Julia

function code(x, eps)
	return fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(Float64(sin(eps) / -1.0) * tan(Float64(eps / 2.0)))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sin[eps], $MachinePrecision] / -1.0), $MachinePrecision] * N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. cos-sum 61.1%

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

   2. cancel-sign-sub-inv 61.1%

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

   3. fma-def 61.1%

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

3. Applied egg-rr 61.1%

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

4. Taylor expanded in x around inf 61.1%

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

   1. associate--l+ 90.6%

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

   2. *-commutative 90.6%

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

   3. neg-mul-1 90.6%

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

   4. distribute-rgt-neg-in 90.6%

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

   5. fma-def 90.6%

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

   6. *-rgt-identity 90.6%

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

   7. distribute-lft-out-- 90.6%

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

   8. sub-neg 90.6%

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

   9. metadata-eval 90.6%

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

   10. +-commutative 90.6%

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

6. Simplified 90.6%

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

   1. flip-+ 90.2%

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

   2. metadata-eval 90.2%

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

   3. 1-sub-cos 99.1%

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

   4. pow2 99.1%

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

8. Applied egg-rr 99.1%

   \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}}\right) \]
9. Step-by-step derivation

   1. sub-neg 99.1%

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

   2. metadata-eval 99.1%

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

   3. distribute-neg-in 99.1%

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

   4. +-commutative 99.1%

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

   5. metadata-eval 99.1%

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

   6. sub-neg 99.1%

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

   7. unpow2 99.1%

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

   8. neg-mul-1 99.1%

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

   9. times-frac 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

   13. hang-0p-tan 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

    \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right) \]

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. cos-sum 61.1%

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

   2. cancel-sign-sub-inv 61.1%

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

   3. fma-def 61.1%

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

3. Applied egg-rr 61.1%

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

4. Taylor expanded in x around inf 61.1%

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

   1. associate--l+ 90.6%

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

   2. *-commutative 90.6%

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

   3. neg-mul-1 90.6%

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

   4. distribute-rgt-neg-in 90.6%

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

   5. fma-def 90.6%

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

   6. *-rgt-identity 90.6%

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

   7. distribute-lft-out-- 90.6%

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

   8. sub-neg 90.6%

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

   9. metadata-eval 90.6%

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

   10. +-commutative 90.6%

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

6. Simplified 90.6%

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

   1. flip-+ 90.2%

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

   2. metadata-eval 90.2%

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

   3. 1-sub-cos 99.1%

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

   4. pow2 99.1%

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

8. Applied egg-rr 99.1%

   \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}}\right) \]
9. Step-by-step derivation

   1. sub-neg 99.1%

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

   2. metadata-eval 99.1%

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

   3. distribute-neg-in 99.1%

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

   4. +-commutative 99.1%

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

   5. metadata-eval 99.1%

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

   6. sub-neg 99.1%

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

   7. unpow2 99.1%

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

   8. neg-mul-1 99.1%

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

   9. times-frac 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

   13. hang-0p-tan 99.5%

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

10. Simplified 99.5%

    \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)}\right) \]
11. Step-by-step derivation

    1. fma-udef 99.5%

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

    2. div-inv 99.5%

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

    3. metadata-eval 99.5%

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

    4. div-inv 99.5%

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

    5. metadata-eval 99.5%

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

12. Applied egg-rr 99.5%

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

    1. distribute-rgt-neg-in 99.5%

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

    2. +-commutative 99.5%

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

    3. associate-*r* 99.5%

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

    4. fma-def 99.5%

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

    5. *-commutative 99.5%

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

    6. neg-mul-1 99.5%

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

    7. *-commutative 99.5%

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

    8. distribute-lft-neg-in 99.5%

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

    9. *-commutative 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. cos-sum 61.1%

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

   2. cancel-sign-sub-inv 61.1%

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

   3. fma-def 61.1%

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

3. Applied egg-rr 61.1%

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

4. Taylor expanded in x around inf 61.1%

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

   1. associate--l+ 90.6%

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

   2. *-commutative 90.6%

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

   3. neg-mul-1 90.6%

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

   4. distribute-rgt-neg-in 90.6%

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

   5. fma-def 90.6%

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

   6. *-rgt-identity 90.6%

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

   7. distribute-lft-out-- 90.6%

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

   8. sub-neg 90.6%

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

   9. metadata-eval 90.6%

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

   10. +-commutative 90.6%

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

6. Simplified 90.6%

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

   1. flip-+ 90.2%

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

   2. metadata-eval 90.2%

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

   3. 1-sub-cos 99.1%

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

   4. pow2 99.1%

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

8. Applied egg-rr 99.1%

   \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon}}\right) \]
9. Step-by-step derivation

   1. sub-neg 99.1%

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

   2. metadata-eval 99.1%

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

   3. distribute-neg-in 99.1%

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

   4. +-commutative 99.1%

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

   5. metadata-eval 99.1%

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

   6. sub-neg 99.1%

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

   7. unpow2 99.1%

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

   8. neg-mul-1 99.1%

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

   9. times-frac 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

   13. hang-0p-tan 99.5%

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

10. Simplified 99.5%

    \[\leadsto \mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)}\right) \]
11. Step-by-step derivation

    1. fma-udef 99.5%

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

    2. div-inv 99.5%

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

    3. metadata-eval 99.5%

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

    4. div-inv 99.5%

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

    5. metadata-eval 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -5 \cdot 10^{-12}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ x eps)) (cos x)) -5e-12)
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))
   (- (* -0.5 (* eps (* eps (cos x)))) (* (sin x) eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((x + eps)) - cos(x)) <= -5e-12) {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (sin(x) * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((x + eps)) - cos(x)) <= (-5d-12)) then
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (sin(x) * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((x + eps)) - Math.cos(x)) <= -5e-12) {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (Math.sin(x) * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((x + eps)) - math.cos(x)) <= -5e-12:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (math.sin(x) * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(x + eps)) - cos(x)) <= -5e-12)
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(sin(x) * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((x + eps)) - cos(x)) <= -5e-12)
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (sin(x) * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -5e-12], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -4.9999999999999997e-12

   1. Initial program 81.4%

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

      1. diff-cos 82.6%

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

      2. div-inv 82.6%

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

      3. metadata-eval 82.6%

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

      4. div-inv 82.6%

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

      5. +-commutative 82.6%

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

      6. metadata-eval 82.6%

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

   3. Applied egg-rr 82.6%

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

      1. *-commutative 82.6%

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

      2. +-commutative 82.6%

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

      3. associate--l+ 82.6%

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

      4. +-inverses 82.6%

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

      5. distribute-lft-in 82.6%

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

      6. metadata-eval 82.6%

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

      7. *-commutative 82.6%

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

      8. +-commutative 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in x around 0 82.6%

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

   if -4.9999999999999997e-12 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 14.0%

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

   2. Taylor expanded in eps around 0 71.9%

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

      1. mul-1-neg 71.9%

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

      2. unsub-neg 71.9%

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

      3. unpow2 71.9%

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

      4. associate-*l* 71.9%

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

   4. Simplified 71.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(x + \varepsilon\right) - \cos x \leq -5 \cdot 10^{-12}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00015d0)) .or. (.not. (eps <= 0.000135d0))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(x) * sin(eps))
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) + (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) - eps))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

      1. cos-sum 99.0%

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

      2. cancel-sign-sub-inv 99.0%

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

      3. fma-def 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in x around inf 99.0%

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

      1. associate--l+ 99.1%

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

      2. *-commutative 99.1%

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

      3. neg-mul-1 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. fma-def 99.1%

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

      6. *-rgt-identity 99.1%

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

      7. distribute-lft-out-- 99.1%

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

      8. sub-neg 99.1%

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

      9. metadata-eval 99.1%

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

      10. +-commutative 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in x around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

      4. +-commutative 99.1%

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

      5. mul-1-neg 99.1%

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

      6. unsub-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. *-commutative 99.1%

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

   9. Simplified 99.1%

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

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

   1. Initial program 17.3%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-+l+ 99.4%

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

      3. unpow2 99.4%

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

      4. associate-*l* 99.4%

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

      5. associate-*r* 99.4%

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

      6. associate-*r* 99.4%

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

      7. distribute-rgt-out 99.4%

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

      8. mul-1-neg 99.4%

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

   4. Simplified 99.4%

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

4. Final simplification 99.3%

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

Alternative 6: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := -2 \cdot \left(\sin x \cdot t_0\right)\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-111}:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-27}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (* -2.0 (* (sin x) t_0))))
   (if (<= x -0.03)
     t_1
     (if (<= x -7e-111)
       (- (+ -1.0 (cos eps)) (* x (sin eps)))
       (if (<= x 8.8e-27) (* -2.0 (pow t_0 2.0)) t_1)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = -2.0 * (sin(x) * t_0);
	double tmp;
	if (x <= -0.03) {
		tmp = t_1;
	} else if (x <= -7e-111) {
		tmp = (-1.0 + cos(eps)) - (x * sin(eps));
	} else if (x <= 8.8e-27) {
		tmp = -2.0 * pow(t_0, 2.0);
	} else {
		tmp = 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 = sin((eps * 0.5d0))
    t_1 = (-2.0d0) * (sin(x) * t_0)
    if (x <= (-0.03d0)) then
        tmp = t_1
    else if (x <= (-7d-111)) then
        tmp = ((-1.0d0) + cos(eps)) - (x * sin(eps))
    else if (x <= 8.8d-27) then
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double t_1 = -2.0 * (Math.sin(x) * t_0);
	double tmp;
	if (x <= -0.03) {
		tmp = t_1;
	} else if (x <= -7e-111) {
		tmp = (-1.0 + Math.cos(eps)) - (x * Math.sin(eps));
	} else if (x <= 8.8e-27) {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	t_1 = -2.0 * (math.sin(x) * t_0)
	tmp = 0
	if x <= -0.03:
		tmp = t_1
	elif x <= -7e-111:
		tmp = (-1.0 + math.cos(eps)) - (x * math.sin(eps))
	elif x <= 8.8e-27:
		tmp = -2.0 * math.pow(t_0, 2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(-2.0 * Float64(sin(x) * t_0))
	tmp = 0.0
	if (x <= -0.03)
		tmp = t_1;
	elseif (x <= -7e-111)
		tmp = Float64(Float64(-1.0 + cos(eps)) - Float64(x * sin(eps)));
	elseif (x <= 8.8e-27)
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	t_1 = -2.0 * (sin(x) * t_0);
	tmp = 0.0;
	if (x <= -0.03)
		tmp = t_1;
	elseif (x <= -7e-111)
		tmp = (-1.0 + cos(eps)) - (x * sin(eps));
	elseif (x <= 8.8e-27)
		tmp = -2.0 * (t_0 ^ 2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.03], t$95$1, If[LessEqual[x, -7e-111], N[(N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(x * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-27], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.029999999999999999 or 8.79999999999999948e-27 < x

   1. Initial program 8.1%

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

      1. diff-cos 7.2%

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

      2. div-inv 7.2%

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

      3. metadata-eval 7.2%

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

      4. div-inv 7.2%

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

      5. +-commutative 7.2%

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

      6. metadata-eval 7.2%

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

   3. Applied egg-rr 7.2%

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

      1. *-commutative 7.2%

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

      2. +-commutative 7.2%

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

      3. associate--l+ 57.0%

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

      4. +-inverses 57.0%

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

      5. distribute-lft-in 57.0%

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

      6. metadata-eval 57.0%

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

      7. *-commutative 57.0%

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

      8. +-commutative 57.0%

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

   5. Simplified 57.0%

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

      1. add-exp-log 23.0%

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

      2. *-commutative 23.0%

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

      3. *-commutative 23.0%

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

      4. +-commutative 23.0%

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

      5. +-rgt-identity 23.0%

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

   7. Applied egg-rr 23.0%

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

      1. add-exp-log 57.0%

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

      2. +-commutative 57.0%

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

      3. *-commutative 57.0%

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

   9. Applied egg-rr 57.0%

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

   10. Taylor expanded in eps around 0 56.9%

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

   if -0.029999999999999999 < x < -7.0000000000000001e-111

   1. Initial program 38.1%

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

   2. Taylor expanded in x around 0 41.4%

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

      1. sub-neg 41.4%

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

      2. metadata-eval 41.4%

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

      3. +-commutative 41.4%

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

      4. associate-+r+ 91.5%

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

      5. +-commutative 91.5%

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

      6. +-commutative 91.5%

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

      7. mul-1-neg 91.5%

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

      8. unsub-neg 91.5%

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

      9. *-commutative 91.5%

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

      10. unpow2 91.5%

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

      11. associate-*l* 91.5%

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

      12. distribute-lft-out-- 91.5%

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

   4. Simplified 91.5%

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

   5. Taylor expanded in x around 0 91.0%

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

      1. associate-*r* 91.0%

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

      2. mul-1-neg 91.0%

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

   7. Simplified 91.0%

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

   if -7.0000000000000001e-111 < x < 8.79999999999999948e-27

   1. Initial program 75.5%

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

      1. diff-cos 96.0%

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

      2. div-inv 96.0%

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

      3. metadata-eval 96.0%

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

      4. div-inv 96.0%

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

      5. +-commutative 96.0%

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

      6. metadata-eval 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. *-commutative 96.0%

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

      2. +-commutative 96.0%

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

      3. associate--l+ 99.6%

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

      4. +-inverses 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 95.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-111}:\\ \;\;\;\;\left(-1 + \cos \varepsilon\right) - x \cdot \sin \varepsilon\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-27}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 76.3% 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 37.2%

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

   1. diff-cos 45.3%

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

   2. div-inv 45.3%

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

   3. metadata-eval 45.3%

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

   4. div-inv 45.3%

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

   5. +-commutative 45.3%

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

   6. metadata-eval 45.3%

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

3. Applied egg-rr 45.3%

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

   1. *-commutative 45.3%

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

   2. +-commutative 45.3%

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

   3. associate--l+ 76.5%

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

   4. +-inverses 76.5%

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

   5. distribute-lft-in 76.5%

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

   6. metadata-eval 76.5%

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

   7. *-commutative 76.5%

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

   8. +-commutative 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in x around -inf 76.4%

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

7. Final simplification 76.4%

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

Alternative 8: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return sin((eps * 0.5)) * (-2.0 * 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 = sin((eps * 0.5d0)) * ((-2.0d0) * sin((0.5d0 * (eps - (x * (-2.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. diff-cos 45.3%

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

   2. div-inv 45.3%

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

   3. metadata-eval 45.3%

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

   4. div-inv 45.3%

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

   5. +-commutative 45.3%

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

   6. metadata-eval 45.3%

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

3. Applied egg-rr 45.3%

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

   1. *-commutative 45.3%

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

   2. +-commutative 45.3%

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

   3. associate--l+ 76.5%

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

   4. +-inverses 76.5%

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

   5. distribute-lft-in 76.5%

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

   6. metadata-eval 76.5%

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

   7. *-commutative 76.5%

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

   8. +-commutative 76.5%

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

5. Simplified 76.5%

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

   1. add-exp-log 20.1%

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

   2. *-commutative 20.1%

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

   3. *-commutative 20.1%

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

   4. +-commutative 20.1%

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

   5. +-rgt-identity 20.1%

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

7. Applied egg-rr 20.1%

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

8. Taylor expanded in x around inf 76.4%

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

   1. *-commutative 76.4%

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

   2. metadata-eval 76.4%

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

   3. cancel-sign-sub-inv 76.4%

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

   4. associate-*r* 76.5%

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

10. Simplified 76.5%

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

11. Final simplification 76.5%

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

Alternative 9: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.2%

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

   1. diff-cos 45.3%

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

   2. div-inv 45.3%

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

   3. metadata-eval 45.3%

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

   4. div-inv 45.3%

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

   5. +-commutative 45.3%

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

   6. metadata-eval 45.3%

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

3. Applied egg-rr 45.3%

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

   1. *-commutative 45.3%

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

   2. +-commutative 45.3%

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

   3. associate--l+ 76.5%

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

   4. +-inverses 76.5%

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

   5. distribute-lft-in 76.5%

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

   6. metadata-eval 76.5%

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

   7. *-commutative 76.5%

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

   8. +-commutative 76.5%

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

5. Simplified 76.5%

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

   1. add-exp-log 20.1%

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

   2. *-commutative 20.1%

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

   3. *-commutative 20.1%

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

   4. +-commutative 20.1%

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

   5. +-rgt-identity 20.1%

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

7. Applied egg-rr 20.1%

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

   1. add-exp-log 76.5%

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

   2. +-commutative 76.5%

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

   3. *-commutative 76.5%

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

9. Applied egg-rr 76.5%

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

10. Final simplification 76.5%

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

Alternative 10: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-92} \lor \neg \left(x \leq 8.8 \cdot 10^{-27}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (if (or (<= x -8.5e-92) (not (<= x 8.8e-27)))
     (* -2.0 (* (sin x) t_0))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double tmp;
	if ((x <= -8.5e-92) || !(x <= 8.8e-27)) {
		tmp = -2.0 * (sin(x) * t_0);
	} else {
		tmp = -2.0 * pow(t_0, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps * 0.5d0))
    if ((x <= (-8.5d-92)) .or. (.not. (x <= 8.8d-27))) then
        tmp = (-2.0d0) * (sin(x) * t_0)
    else
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	double tmp;
	if ((x <= -8.5e-92) || !(x <= 8.8e-27)) {
		tmp = -2.0 * (Math.sin(x) * t_0);
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	tmp = 0
	if (x <= -8.5e-92) or not (x <= 8.8e-27):
		tmp = -2.0 * (math.sin(x) * t_0)
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	tmp = 0.0
	if ((x <= -8.5e-92) || !(x <= 8.8e-27))
		tmp = Float64(-2.0 * Float64(sin(x) * t_0));
	else
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 0.0;
	if ((x <= -8.5e-92) || ~((x <= 8.8e-27)))
		tmp = -2.0 * (sin(x) * t_0);
	else
		tmp = -2.0 * (t_0 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -8.5e-92], N[Not[LessEqual[x, 8.8e-27]], $MachinePrecision]], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000067e-92 or 8.79999999999999948e-27 < x

   1. Initial program 10.3%

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

      1. diff-cos 10.1%

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

      2. div-inv 10.1%

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

      3. metadata-eval 10.1%

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

      4. div-inv 10.1%

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

      5. +-commutative 10.1%

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

      6. metadata-eval 10.1%

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

   3. Applied egg-rr 10.1%

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

      1. *-commutative 10.1%

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

      2. +-commutative 10.1%

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

      3. associate--l+ 60.0%

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

      4. +-inverses 60.0%

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

      5. distribute-lft-in 60.0%

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

      6. metadata-eval 60.0%

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

      7. *-commutative 60.0%

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

      8. +-commutative 60.0%

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

   5. Simplified 60.0%

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

      1. add-exp-log 24.0%

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

      2. *-commutative 24.0%

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

      3. *-commutative 24.0%

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

      4. +-commutative 24.0%

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

      5. +-rgt-identity 24.0%

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

   7. Applied egg-rr 24.0%

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

      1. add-exp-log 60.0%

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

      2. +-commutative 60.0%

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

      3. *-commutative 60.0%

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

   9. Applied egg-rr 60.0%

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

   10. Taylor expanded in eps around 0 56.8%

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

   if -8.50000000000000067e-92 < x < 8.79999999999999948e-27

   1. Initial program 74.7%

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

      1. diff-cos 94.4%

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

      2. div-inv 94.4%

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

      3. metadata-eval 94.4%

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

      4. div-inv 94.4%

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

      5. +-commutative 94.4%

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

      6. metadata-eval 94.4%

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

   3. Applied egg-rr 94.4%

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

      1. *-commutative 94.4%

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

      2. +-commutative 94.4%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 93.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-92} \lor \neg \left(x \leq 8.8 \cdot 10^{-27}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 11: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-92} \lor \neg \left(x \leq 1.2 \cdot 10^{-21}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -8.5e-92) (not (<= x 1.2e-21)))
   (* (sin x) (- eps))
   (* -2.0 (pow (sin (* eps 0.5)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -8.5e-92) || !(x <= 1.2e-21)) {
		tmp = sin(x) * -eps;
	} else {
		tmp = -2.0 * pow(sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-8.5d-92)) .or. (.not. (x <= 1.2d-21))) then
        tmp = sin(x) * -eps
    else
        tmp = (-2.0d0) * (sin((eps * 0.5d0)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -8.5e-92) || !(x <= 1.2e-21)) {
		tmp = Math.sin(x) * -eps;
	} else {
		tmp = -2.0 * Math.pow(Math.sin((eps * 0.5)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -8.5e-92) or not (x <= 1.2e-21):
		tmp = math.sin(x) * -eps
	else:
		tmp = -2.0 * math.pow(math.sin((eps * 0.5)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -8.5e-92) || !(x <= 1.2e-21))
		tmp = Float64(sin(x) * Float64(-eps));
	else
		tmp = Float64(-2.0 * (sin(Float64(eps * 0.5)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -8.5e-92) || ~((x <= 1.2e-21)))
		tmp = sin(x) * -eps;
	else
		tmp = -2.0 * (sin((eps * 0.5)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -8.5e-92], N[Not[LessEqual[x, 1.2e-21]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000067e-92 or 1.2e-21 < x

   1. Initial program 10.3%

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

   2. Taylor expanded in eps around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. mul-1-neg 53.5%

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

   4. Simplified 53.5%

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

   if -8.50000000000000067e-92 < x < 1.2e-21

   1. Initial program 74.7%

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

      1. diff-cos 94.4%

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

      2. div-inv 94.4%

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

      3. metadata-eval 94.4%

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

      4. div-inv 94.4%

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

      5. +-commutative 94.4%

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

      6. metadata-eval 94.4%

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

   3. Applied egg-rr 94.4%

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

      1. *-commutative 94.4%

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

      2. +-commutative 94.4%

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

      3. associate--l+ 99.5%

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

      4. +-inverses 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 93.7%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-92} \lor \neg \left(x \leq 1.2 \cdot 10^{-21}\right):\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2}\\ \end{array} \]

Alternative 12: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-11} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.7e-11) (not (<= eps 1.9e-6)))
   (- (cos eps) (cos x))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.7e-11) || !(eps <= 1.9e-6)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.7d-11)) .or. (.not. (eps <= 1.9d-6))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.7e-11) || !(eps <= 1.9e-6)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.7e-11) or not (eps <= 1.9e-6):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.7e-11) || !(eps <= 1.9e-6))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.7e-11) || ~((eps <= 1.9e-6)))
		tmp = cos(eps) - cos(x);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.7e-11], N[Not[LessEqual[eps, 1.9e-6]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.70000000000000005e-11 or 1.9e-6 < eps

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0 56.2%

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

   if -2.70000000000000005e-11 < eps < 1.9e-6

   1. Initial program 17.0%

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

   2. Taylor expanded in eps around 0 82.9%

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

      1. associate-*r* 82.9%

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

      2. mul-1-neg 82.9%

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

   4. Simplified 82.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-11} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 13: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-11} \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.7e-11) (not (<= eps 2.7e-6)))
   (+ -1.0 (cos eps))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.7e-11) || !(eps <= 2.7e-6)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.7d-11)) .or. (.not. (eps <= 2.7d-6))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.7e-11) || !(eps <= 2.7e-6)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.7e-11) or not (eps <= 2.7e-6):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.7e-11) || !(eps <= 2.7e-6))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.7e-11) || ~((eps <= 2.7e-6)))
		tmp = -1.0 + cos(eps);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.7e-11], N[Not[LessEqual[eps, 2.7e-6]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.70000000000000005e-11 or 2.69999999999999998e-6 < eps

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0 55.0%

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

   if -2.70000000000000005e-11 < eps < 2.69999999999999998e-6

   1. Initial program 17.0%

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

   2. Taylor expanded in eps around 0 82.9%

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

      1. associate-*r* 82.9%

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

      2. mul-1-neg 82.9%

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

   4. Simplified 82.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-11} \lor \neg \left(\varepsilon \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 14: 46.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00015d0)) .or. (.not. (eps <= 0.000135d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = (-0.5d0) * (eps * eps)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

   2. Taylor expanded in x around 0 55.8%

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

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

   1. Initial program 17.3%

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

   2. Taylor expanded in x around 0 17.3%

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

   3. Taylor expanded in eps around 0 34.2%

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

      1. unpow2 34.2%

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

   5. Simplified 34.2%

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

4. Final simplification 45.6%

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

Alternative 15: 21.0% 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 37.2%

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

2. Taylor expanded in x around 0 37.6%

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

3. Taylor expanded in eps around 0 18.1%

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

   1. unpow2 18.1%

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

5. Simplified 18.1%

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

6. Final simplification 18.1%

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

Reproduce

herbie shell --seed 2023273 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))