2sin (example 3.3)

Percentage Accurate: 62.5% → 100.0%

Time: 15.1s

Alternatives: 10

Speedup: 205.0×

Specification

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

Math

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

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: 62.5% 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: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. sin-sum 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in x around inf 62.8%

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

   1. sub-neg 62.8%

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

   2. +-commutative 62.8%

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

   3. *-commutative 62.8%

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

   4. *-commutative 62.8%

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

   5. associate-+r+ 99.9%

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

   6. fma-define 99.9%

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

   7. +-commutative 99.9%

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

   8. neg-mul-1 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-out 99.9%

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

7. Simplified 99.9%

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

   1. flip-+ 99.9%

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

   2. frac-2neg 99.9%

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

   3. metadata-eval 99.9%

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

   4. 1-sub-cos 100.0%

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

   5. pow2 100.0%

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

9. Applied egg-rr 100.0%

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

    1. distribute-frac-neg 100.0%

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

    2. unpow2 100.0%

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

    3. associate-/l* 100.0%

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

    4. neg-sub0 100.0%

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

    5. associate--r- 100.0%

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

    6. metadata-eval 100.0%

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

    7. hang-0p-tan 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. sin-sum 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in x around inf 62.8%

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

   1. sub-neg 62.8%

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

   2. +-commutative 62.8%

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

   3. *-commutative 62.8%

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

   4. *-commutative 62.8%

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

   5. associate-+r+ 99.9%

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

   6. fma-define 99.9%

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

   7. +-commutative 99.9%

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

   8. neg-mul-1 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-out 99.9%

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

7. Simplified 99.9%

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

   1. flip-+ 99.9%

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

   2. frac-2neg 99.9%

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

   3. metadata-eval 99.9%

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

   4. 1-sub-cos 100.0%

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

   5. pow2 100.0%

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

9. Applied egg-rr 100.0%

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

    1. distribute-frac-neg 100.0%

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

    2. unpow2 100.0%

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

    3. associate-/l* 100.0%

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

    4. neg-sub0 100.0%

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

    5. associate--r- 100.0%

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

    6. metadata-eval 100.0%

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

    7. hang-0p-tan 100.0%

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

11. Simplified 100.0%

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

    1. distribute-rgt-neg-out 100.0%

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

    2. fmm-undef 100.0%

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

    3. div-inv 100.0%

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

    4. metadata-eval 100.0%

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

13. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x - \sin x \cdot \left(\sin \varepsilon \cdot \tan \left(\varepsilon \cdot 0.5\right)\right)} \]
14. Add Preprocessing

Alternative 3: 99.7% 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 62.6%

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

   1. sin-sum 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in x around inf 62.8%

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

   1. sub-neg 62.8%

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

   2. +-commutative 62.8%

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

   3. *-commutative 62.8%

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

   4. *-commutative 62.8%

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

   5. associate-+r+ 99.9%

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

   6. fma-define 99.9%

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

   7. +-commutative 99.9%

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

   8. neg-mul-1 99.9%

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

   9. *-commutative 99.9%

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

   10. distribute-rgt-out 99.9%

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

7. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

   2. +-commutative 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. diff-sin 62.7%

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

   2. div-inv 62.7%

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

   3. associate--l+ 62.7%

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

   4. metadata-eval 62.7%

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

   5. div-inv 62.7%

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

   6. +-commutative 62.7%

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

   7. associate-+l+ 62.7%

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

   8. metadata-eval 62.7%

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

4. Applied egg-rr 62.7%

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

   1. associate-*r* 62.7%

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

   2. *-commutative 62.7%

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

   3. *-commutative 62.7%

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

   4. +-commutative 62.7%

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

   5. count-2 62.7%

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

   6. fma-define 62.7%

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

   7. associate-+r- 62.7%

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

   8. +-commutative 62.7%

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

   9. associate--l+ 99.9%

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

   10. +-inverses 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in x around inf 99.9%

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

   1. *-commutative 99.9%

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

   2. metadata-eval 99.9%

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

   3. cancel-sign-sub-inv 99.9%

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

   4. cancel-sign-sub-inv 99.9%

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

   5. metadata-eval 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-undefine 99.9%

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

   8. associate-*r* 99.9%

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

   9. fma-undefine 99.9%

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

   10. +-commutative 99.9%

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

   11. distribute-lft-in 99.9%

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

   12. associate-*r* 99.9%

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

   13. metadata-eval 99.9%

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

   14. *-lft-identity 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.3%

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

Alternative 6: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
4. Add Preprocessing

Alternative 7: 98.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.6%

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

   1. distribute-lft-out 98.6%

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

   2. unpow2 98.6%

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

   3. distribute-lft-out 98.6%

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

   4. +-commutative 98.6%

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

6. Simplified 98.6%

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

Alternative 8: 98.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.6%

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

   1. distribute-lft-out 98.6%

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

   2. unpow2 98.6%

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

   3. distribute-lft-out 98.6%

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

   4. +-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in eps around 0 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r* 98.5%

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

9. Simplified 98.5%

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

Alternative 9: 97.8% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + -0.5 \cdot \left(\varepsilon \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ 1.0 (* -0.5 (* eps x)))))

C

double code(double x, double eps) {
	return eps * (1.0 + (-0.5 * (eps * x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((-0.5d0) * (eps * x)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (-0.5 * (eps * x)));
}

Python

def code(x, eps):
	return eps * (1.0 + (-0.5 * (eps * x)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(-0.5 * Float64(eps * x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (-0.5 * (eps * x)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(-0.5 * N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\varepsilon \cdot x\right)\right)} \]
5. Add Preprocessing

Alternative 10: 97.8% 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 62.6%

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

3. Taylor expanded in eps around 0 99.3%

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

4. Taylor expanded in x around 0 98.3%

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

5. Taylor expanded in eps around 0 98.3%

   \[\leadsto \color{blue}{\varepsilon} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * N[Cos[N[(x + N[(eps / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Developer Target 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024180 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (+ (* (sin x) (- (cos eps) 1)) (* (cos x) (sin eps))))

  :alt
  (! :herbie-platform default (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

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