2sin (example 3.3)

Percentage Accurate: 41.9% → 99.4%

Time: 16.0s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. sin-sum 69.1%

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

   2. associate--l+ 69.1%

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

3. Applied egg-rr 69.1%

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

   1. associate-+r- 69.1%

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

   2. +-commutative 69.1%

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

   3. associate-+r- 99.2%

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

   4. *-commutative 99.2%

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

   5. sub-neg 99.2%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.3%

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

5. Simplified 99.3%

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

   1. flip-+ 99.2%

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

   2. frac-2neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. sub-1-cos 99.6%

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

   5. pow2 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

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

   1. remove-double-neg 99.6%

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

   2. distribute-neg-in 99.6%

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

   3. metadata-eval 99.6%

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

   4. +-commutative 99.6%

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

   5. sub-neg 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x - \frac{{\sin \varepsilon}^{2}}{\frac{\cos \varepsilon + 1}{\sin x}} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (* (sin eps) (cos x))
  (/ (pow (sin eps) 2.0) (/ (+ (cos eps) 1.0) (sin x)))))

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) - ((sin(eps) ** 2.0d0) / ((cos(eps) + 1.0d0) / sin(x)))
end function

Java

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) - (Math.pow(Math.sin(eps), 2.0) / ((Math.cos(eps) + 1.0) / Math.sin(x)));
}

Python

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) - (math.pow(math.sin(eps), 2.0) / ((math.cos(eps) + 1.0) / math.sin(x)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) - ((sin(eps) ^ 2.0) / ((cos(eps) + 1.0) / sin(x)));
end

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. sin-sum 69.1%

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

   2. associate--l+ 69.1%

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

3. Applied egg-rr 69.1%

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

   1. associate-+r- 69.1%

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

   2. +-commutative 69.1%

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

   3. associate-+r- 99.2%

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

   4. *-commutative 99.2%

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

   5. sub-neg 99.2%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.3%

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

5. Simplified 99.3%

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

   1. fma-udef 99.3%

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

   2. *-commutative 99.3%

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

   3. distribute-lft-in 99.2%

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

   4. +-commutative 99.2%

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

   5. *-commutative 99.2%

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

   6. mul-1-neg 99.2%

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

   7. associate-+l+ 69.1%

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

   8. sub-neg 69.1%

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

   9. associate-+l- 99.2%

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

   10. *-commutative 99.2%

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

   11. *-un-lft-identity 99.2%

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

   12. metadata-eval 99.2%

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

   13. *-commutative 99.2%

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

   14. distribute-rgt-out-- 99.3%

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

   15. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

   1. flip-- 99.2%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{1 \cdot 1 - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon}} \]

   2. metadata-eval 99.2%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{1} - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon} \]

   3. metadata-eval 99.2%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \frac{\color{blue}{-1 \cdot -1} - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon} \]

   4. associate-*r/ 99.2%

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

   5. metadata-eval 99.2%

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

   6. 1-sub-cos 99.6%

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

   7. pow2 99.6%

      \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\sin x \cdot \color{blue}{{\sin \varepsilon}^{2}}}{1 + \cos \varepsilon} \]

   8. +-commutative 99.6%

      \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\sin x \cdot {\sin \varepsilon}^{2}}{\color{blue}{\cos \varepsilon + 1}} \]

9. Applied egg-rr 99.6%

   \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{\sin x \cdot {\sin \varepsilon}^{2}}{\cos \varepsilon + 1}} \]
10. Step-by-step derivation

    1. *-commutative 99.6%

       \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \sin x}}{\cos \varepsilon + 1} \]

    2. associate-/l* 99.6%

       \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{\cos \varepsilon + 1}{\sin x}}} \]

    3. +-commutative 99.6%

       \[\leadsto \sin \varepsilon \cdot \cos x - \frac{{\sin \varepsilon}^{2}}{\frac{\color{blue}{1 + \cos \varepsilon}}{\sin x}} \]

11. Simplified 99.6%

    \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

12. Final simplification 99.6%

    \[\leadsto \sin \varepsilon \cdot \cos x - \frac{{\sin \varepsilon}^{2}}{\frac{\cos \varepsilon + 1}{\sin x}} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (sin eps) (cos x) (* (sin x) (+ -1.0 (cos eps)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. sin-sum 69.1%

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

   2. associate--l+ 69.1%

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

3. Applied egg-rr 69.1%

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

   1. associate-+r- 69.1%

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

   2. +-commutative 69.1%

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

   3. associate-+r- 99.2%

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

   4. *-commutative 99.2%

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

   5. sub-neg 99.2%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (sin x) (+ -1.0 (cos eps)) (* (sin eps) (cos x))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. sin-sum 69.1%

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

   2. associate--l+ 69.1%

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

3. Applied egg-rr 69.1%

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

   1. associate-+r- 69.1%

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

   2. +-commutative 69.1%

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

   3. associate-+r- 99.2%

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

   4. *-commutative 99.2%

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

   5. sub-neg 99.2%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.3%

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

5. Simplified 99.3%

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

   1. fma-udef 99.3%

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

   2. *-commutative 99.3%

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

   3. distribute-lft-in 99.2%

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

   4. +-commutative 99.2%

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

   5. *-commutative 99.2%

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

   6. mul-1-neg 99.2%

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

   7. associate-+l+ 69.1%

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

   8. sub-neg 69.1%

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

   9. associate-+l- 99.2%

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

   10. *-commutative 99.2%

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

   11. *-un-lft-identity 99.2%

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

   12. metadata-eval 99.2%

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

   13. *-commutative 99.2%

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

   14. distribute-rgt-out-- 99.3%

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

   15. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

8. Taylor expanded in eps around inf 99.3%

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

   1. cancel-sign-sub-inv 99.3%

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

   2. sub-neg 99.3%

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

   3. distribute-lft-in 99.2%

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

   4. *-commutative 99.2%

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

   5. *-rgt-identity 99.2%

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

   6. associate-+l+ 69.1%

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

   7. distribute-rgt-neg-out 69.1%

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

   8. distribute-lft-neg-out 69.1%

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

   9. remove-double-neg 69.1%

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

   10. associate-+r+ 99.2%

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

   11. neg-mul-1 99.2%

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

   12. distribute-rgt-in 99.3%

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

   13. +-commutative 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(-1 + \cos \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (sin eps) (cos x)) (* (sin x) (+ -1.0 (cos eps)))))

C

double code(double x, double eps) {
	return (sin(eps) * cos(x)) + (sin(x) * (-1.0 + cos(eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) + (sin(x) * ((-1.0d0) + cos(eps)))
end function

Java

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) + (Math.sin(x) * (-1.0 + Math.cos(eps)));
}

Python

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) + (math.sin(x) * (-1.0 + math.cos(eps)))

Julia

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) + Float64(sin(x) * Float64(-1.0 + cos(eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) + (sin(x) * (-1.0 + cos(eps)));
end

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. sin-sum 69.1%

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

   2. associate--l+ 69.1%

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

3. Applied egg-rr 69.1%

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

   1. associate-+r- 69.1%

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

   2. +-commutative 69.1%

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

   3. associate-+r- 99.2%

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

   4. *-commutative 99.2%

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

   5. sub-neg 99.2%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.3%

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

5. Simplified 99.3%

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

   1. fma-udef 99.3%

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

   2. *-commutative 99.3%

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

   3. distribute-lft-in 99.2%

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

   4. +-commutative 99.2%

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

   5. *-commutative 99.2%

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

   6. mul-1-neg 99.2%

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

   7. associate-+l+ 69.1%

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

   8. sub-neg 69.1%

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

   9. associate-+l- 99.2%

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

   10. *-commutative 99.2%

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

   11. *-un-lft-identity 99.2%

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

   12. metadata-eval 99.2%

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

   13. *-commutative 99.2%

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

   14. distribute-rgt-out-- 99.3%

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

   15. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 6: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \left(1 + \left(\sin x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.5e-6)
   (sin eps)
   (if (<= eps 6e-5)
     (* eps (cos x))
     (- (sin (+ eps x)) (+ 1.0 (+ (sin x) -1.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.5e-6) {
		tmp = sin(eps);
	} else if (eps <= 6e-5) {
		tmp = eps * cos(x);
	} else {
		tmp = sin((eps + x)) - (1.0 + (sin(x) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-3.5d-6)) then
        tmp = sin(eps)
    else if (eps <= 6d-5) then
        tmp = eps * cos(x)
    else
        tmp = sin((eps + x)) - (1.0d0 + (sin(x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3.5e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 6e-5) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin((eps + x)) - (1.0 + (Math.sin(x) + -1.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.5e-6:
		tmp = math.sin(eps)
	elif eps <= 6e-5:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin((eps + x)) - (1.0 + (math.sin(x) + -1.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.5e-6)
		tmp = sin(eps);
	elseif (eps <= 6e-5)
		tmp = Float64(eps * cos(x));
	else
		tmp = Float64(sin(Float64(eps + x)) - Float64(1.0 + Float64(sin(x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.5e-6)
		tmp = sin(eps);
	elseif (eps <= 6e-5)
		tmp = eps * cos(x);
	else
		tmp = sin((eps + x)) - (1.0 + (sin(x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.5e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 6e-5], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[(1.0 + N[(N[Sin[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.49999999999999995e-6

   1. Initial program 49.2%

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

   2. Taylor expanded in x around 0 50.6%

      \[\leadsto \color{blue}{\sin \varepsilon} \]

   if -3.49999999999999995e-6 < eps < 6.00000000000000015e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in eps around 0 99.0%

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

   if 6.00000000000000015e-5 < eps

   1. Initial program 54.7%

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

      1. expm1-log1p-u 54.6%

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

      2. expm1-udef 54.7%

         \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x\right)} - 1\right)} \]

      3. associate--r- 54.4%

         \[\leadsto \color{blue}{\left(\sin \left(x + \varepsilon\right) - e^{\mathsf{log1p}\left(\sin x\right)}\right) + 1} \]

   3. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{\left(\sin \left(x + \varepsilon\right) - e^{\mathsf{log1p}\left(\sin x\right)}\right) + 1} \]
   4. Step-by-step derivation

      1. associate-+l- 54.7%

         \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \left(e^{\mathsf{log1p}\left(\sin x\right)} - 1\right)} \]

      2. +-commutative 54.7%

         \[\leadsto \sin \color{blue}{\left(\varepsilon + x\right)} - \left(e^{\mathsf{log1p}\left(\sin x\right)} - 1\right) \]

      3. expm1-def 54.6%

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

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\sin \left(\varepsilon + x\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin x\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-udef 54.7%

         \[\leadsto \sin \left(\varepsilon + x\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x\right)} - 1\right)} \]

      2. metadata-eval 54.7%

         \[\leadsto \sin \left(\varepsilon + x\right) - \left(e^{\mathsf{log1p}\left(\sin x\right)} - \color{blue}{-1 \cdot -1}\right) \]

      3. sub-neg 54.7%

         \[\leadsto \sin \left(\varepsilon + x\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x\right)} + \left(--1 \cdot -1\right)\right)} \]

      4. log1p-udef 54.7%

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

      5. metadata-eval 54.7%

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

      6. add-exp-log 54.7%

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

      7. metadata-eval 54.7%

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

      8. metadata-eval 54.7%

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

      9. associate-+l+ 55.0%

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

      10. metadata-eval 55.0%

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

   7. Applied egg-rr 55.0%

      \[\leadsto \sin \left(\varepsilon + x\right) - \color{blue}{\left(1 + \left(\sin x + -1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \left(1 + \left(\sin x + -1\right)\right)\\ \end{array} \]

Alternative 7: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. diff-sin 43.7%

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

   2. div-inv 43.7%

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

   3. +-commutative 43.7%

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

   4. associate--l+ 75.1%

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

   5. *-un-lft-identity 75.1%

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

   6. *-un-lft-identity 75.1%

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

   7. distribute-rgt-out-- 75.1%

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

   8. metadata-eval 75.1%

      \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot \color{blue}{0}\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   9. metadata-eval 75.1%

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

   10. div-inv 75.1%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

   11. +-commutative 75.1%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\left(\color{blue}{\left(\varepsilon + x\right)} + x\right) \cdot \frac{1}{2}\right)\right) \]

   12. associate-+l+ 75.2%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\color{blue}{\left(\varepsilon + \left(x + x\right)\right)} \cdot \frac{1}{2}\right)\right) \]

   13. count-2 75.2%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\left(\varepsilon + \color{blue}{2 \cdot x}\right) \cdot \frac{1}{2}\right)\right) \]

   14. *-commutative 75.2%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\left(\varepsilon + \color{blue}{x \cdot 2}\right) \cdot \frac{1}{2}\right)\right) \]

   15. metadata-eval 75.2%

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

3. Applied egg-rr 75.2%

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

   1. *-commutative 75.2%

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

   2. associate-*l* 75.2%

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

   3. mul0-rgt 75.2%

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

   4. *-commutative 75.2%

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

5. Simplified 75.2%

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

6. Final simplification 75.2%

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

Alternative 8: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -7.2e-5)
   (sin eps)
   (if (<= eps 1.05e-5) (* eps (cos x)) (- (sin (+ eps x)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -7.2e-5) {
		tmp = sin(eps);
	} else if (eps <= 1.05e-5) {
		tmp = eps * cos(x);
	} else {
		tmp = sin((eps + x)) - sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-7.2d-5)) then
        tmp = sin(eps)
    else if (eps <= 1.05d-5) then
        tmp = eps * cos(x)
    else
        tmp = sin((eps + x)) - sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -7.2e-5) {
		tmp = Math.sin(eps);
	} else if (eps <= 1.05e-5) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin((eps + x)) - Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -7.2e-5:
		tmp = math.sin(eps)
	elif eps <= 1.05e-5:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin((eps + x)) - math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -7.2e-5)
		tmp = sin(eps);
	elseif (eps <= 1.05e-5)
		tmp = Float64(eps * cos(x));
	else
		tmp = Float64(sin(Float64(eps + x)) - sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -7.2e-5)
		tmp = sin(eps);
	elseif (eps <= 1.05e-5)
		tmp = eps * cos(x);
	else
		tmp = sin((eps + x)) - sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -7.2e-5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 1.05e-5], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.20000000000000018e-5

   1. Initial program 49.2%

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

   2. Taylor expanded in x around 0 50.6%

      \[\leadsto \color{blue}{\sin \varepsilon} \]

   if -7.20000000000000018e-5 < eps < 1.04999999999999994e-5

   1. Initial program 36.0%

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

   2. Taylor expanded in eps around 0 99.0%

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

   if 1.04999999999999994e-5 < eps

   1. Initial program 54.7%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \]

Alternative 9: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.000105\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.4e-5) (not (<= eps 0.000105))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.4e-5) || !(eps <= 0.000105)) {
		tmp = sin(eps);
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.4d-5)) .or. (.not. (eps <= 0.000105d0))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.4e-5) || !(eps <= 0.000105)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.4e-5) or not (eps <= 0.000105):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.4e-5) || !(eps <= 0.000105))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.4e-5) || ~((eps <= 0.000105)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.4e-5], N[Not[LessEqual[eps, 0.000105]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.39999999999999998e-5 or 1.05e-4 < eps

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 52.1%

      \[\leadsto \color{blue}{\sin \varepsilon} \]

   if -1.39999999999999998e-5 < eps < 1.05e-4

   1. Initial program 36.0%

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

   2. Taylor expanded in eps around 0 99.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.000105\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 10: 54.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in x around 0 55.9%

   \[\leadsto \color{blue}{\sin \varepsilon} \]

3. Final simplification 55.9%

   \[\leadsto \sin \varepsilon \]

Alternative 11: 29.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 44.2%

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

2. Taylor expanded in eps around 0 49.8%

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

3. Taylor expanded in x around 0 30.8%

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

4. Final simplification 30.8%

   \[\leadsto \varepsilon \]

Developer target: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023305 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

  (- (sin (+ x eps)) (sin x)))