2sin (example 3.3)

Percentage Accurate: 41.8% → 99.6%

Time: 11.2s

Alternatives: 10

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: 41.8% 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: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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) / (-1.0d0)) * tan((eps / 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.1%

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

   1. sin-sum 68.0%

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

   2. associate--l+ 67.9%

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

3. Applied egg-rr 67.9%

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

   1. +-commutative 67.9%

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

   2. sub-neg 67.9%

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

   3. associate-+l+ 99.6%

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

   4. *-commutative 99.6%

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

   5. neg-mul-1 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-out 99.6%

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

   8. +-commutative 99.6%

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

5. Simplified 99.6%

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

   1. add-log-exp 99.7%

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

7. Applied egg-rr 99.7%

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

   1. add-log-exp 99.6%

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

   2. flip-+ 99.5%

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

   3. frac-2neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. sub-1-cos 99.6%

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

   6. pow2 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

9. Applied egg-rr 99.6%

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

    1. remove-double-neg 99.6%

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

    2. unpow2 99.6%

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

    3. neg-mul-1 99.6%

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

    4. times-frac 99.6%

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

    5. +-commutative 99.6%

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

    6. hang-0p-tan 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.1%

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

   1. sin-sum 68.0%

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

   2. associate--l+ 67.9%

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

3. Applied egg-rr 67.9%

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

   1. +-commutative 67.9%

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

   2. sub-neg 67.9%

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

   3. associate-+l+ 99.6%

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

   4. *-commutative 99.6%

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

   5. neg-mul-1 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-out 99.6%

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

   8. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in eps around inf 99.6%

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

   1. +-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. *-commutative 99.6%

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

   4. fma-def 99.7%

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

   5. sub-neg 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.3% 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 46.1%

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

   1. sin-sum 68.0%

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

   2. associate--l+ 67.9%

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

3. Applied egg-rr 67.9%

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

   1. +-commutative 67.9%

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

   2. sub-neg 67.9%

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

   3. associate-+l+ 99.6%

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

   4. *-commutative 99.6%

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

   5. neg-mul-1 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-out 99.6%

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

   8. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0, \sin x, \sin \varepsilon \cdot \cos x\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma 0.0 (sin x) (* (sin eps) (cos x))))

C

double code(double x, double eps) {
	return fma(0.0, sin(x), (sin(eps) * cos(x)));
}

Julia

function code(x, eps)
	return fma(0.0, sin(x), Float64(sin(eps) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(0.0 * N[Sin[x], $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.1%

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

   1. sin-sum 68.0%

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

   2. associate--l+ 67.9%

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

3. Applied egg-rr 67.9%

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

   1. +-commutative 67.9%

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

   2. sub-neg 67.9%

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

   3. associate-+l+ 99.6%

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

   4. *-commutative 99.6%

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

   5. neg-mul-1 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-out 99.6%

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

   8. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in eps around inf 99.6%

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

   1. +-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. *-commutative 99.6%

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

   4. fma-def 99.7%

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

   5. sub-neg 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in eps around 0 79.9%

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

10. Final simplification 79.9%

    \[\leadsto \mathsf{fma}\left(0, \sin x, \sin \varepsilon \cdot \cos x\right) \]

Alternative 5: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.3 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0085:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -5.3e-6)
   (sin eps)
   (if (<= eps 0.0085)
     (+ (* -0.5 (* (sin x) (* eps eps))) (* eps (cos x)))
     (- (sin (+ eps x)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -5.3e-6) {
		tmp = sin(eps);
	} else if (eps <= 0.0085) {
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (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 <= (-5.3d-6)) then
        tmp = sin(eps)
    else if (eps <= 0.0085d0) then
        tmp = ((-0.5d0) * (sin(x) * (eps * eps))) + (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 <= -5.3e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.0085) {
		tmp = (-0.5 * (Math.sin(x) * (eps * eps))) + (eps * Math.cos(x));
	} else {
		tmp = Math.sin((eps + x)) - Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -5.3e-6:
		tmp = math.sin(eps)
	elif eps <= 0.0085:
		tmp = (-0.5 * (math.sin(x) * (eps * eps))) + (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 <= -5.3e-6)
		tmp = sin(eps);
	elseif (eps <= 0.0085)
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * Float64(eps * eps))) + 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 <= -5.3e-6)
		tmp = sin(eps);
	elseif (eps <= 0.0085)
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (eps * cos(x));
	else
		tmp = sin((eps + x)) - sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -5.3e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.0085], N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $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 < -5.3000000000000001e-6

   1. Initial program 66.7%

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

   2. Taylor expanded in x around 0 67.6%

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

   if -5.3000000000000001e-6 < eps < 0.0085000000000000006

   1. Initial program 31.2%

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

   2. Taylor expanded in eps around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. unpow2 99.5%

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

      4. associate-*l* 99.5%

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

      5. *-commutative 99.5%

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

   4. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. associate-*r* 99.5%

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

   6. Applied egg-rr 99.5%

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

   if 0.0085000000000000006 < eps

   1. Initial program 50.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.3 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0085:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \]

Alternative 6: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.1%

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

   1. add-cbrt-cube 40.2%

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

   2. pow3 40.2%

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

3. Applied egg-rr 40.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\sin \left(x + \varepsilon\right) - \sin x\right)}^{3}}} \]
4. Step-by-step derivation

   1. rem-cbrt-cube 46.1%

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

   2. diff-sin 45.4%

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

   3. +-commutative 45.4%

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

   4. +-commutative 45.4%

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

5. Applied egg-rr 45.4%

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

   1. *-commutative 45.4%

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

   2. associate-*l* 45.4%

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

   3. associate--l+ 78.1%

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

   4. +-inverses 78.1%

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

   5. +-commutative 78.1%

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

   6. remove-double-neg 78.1%

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

   7. mul-1-neg 78.1%

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

   8. sub-neg 78.1%

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

   9. neg-sub0 78.1%

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

   10. mul-1-neg 78.1%

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

   11. remove-double-neg 78.1%

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

   12. associate-+l+ 78.1%

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

7. Simplified 78.1%

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

8. Final simplification 78.1%

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

Alternative 7: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00044:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.4e-6)
   (sin eps)
   (if (<= eps 0.00044) (* eps (cos x)) (- (sin (+ eps x)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.4e-6) {
		tmp = sin(eps);
	} else if (eps <= 0.00044) {
		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 <= (-2.4d-6)) then
        tmp = sin(eps)
    else if (eps <= 0.00044d0) 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 <= -2.4e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.00044) {
		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 <= -2.4e-6:
		tmp = math.sin(eps)
	elif eps <= 0.00044:
		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 <= -2.4e-6)
		tmp = sin(eps);
	elseif (eps <= 0.00044)
		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 <= -2.4e-6)
		tmp = sin(eps);
	elseif (eps <= 0.00044)
		tmp = eps * cos(x);
	else
		tmp = sin((eps + x)) - sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.4e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.00044], 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 < -2.3999999999999999e-6

   1. Initial program 66.7%

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

   2. Taylor expanded in x around 0 67.6%

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

   if -2.3999999999999999e-6 < eps < 4.40000000000000016e-4

   1. Initial program 31.2%

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

   2. Taylor expanded in eps around 0 99.3%

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

   if 4.40000000000000016e-4 < eps

   1. Initial program 50.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00044:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \]

Alternative 8: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000255:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.5e-6)
   (sin eps)
   (if (<= eps 0.000255) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.5e-6) {
		tmp = sin(eps);
	} else if (eps <= 0.000255) {
		tmp = eps * cos(x);
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-3.5d-6)) then
        tmp = sin(eps)
    else if (eps <= 0.000255d0) then
        tmp = eps * cos(x)
    else
        tmp = sin(eps)
    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 <= 0.000255) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.5e-6:
		tmp = math.sin(eps)
	elif eps <= 0.000255:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.5e-6)
		tmp = sin(eps);
	elseif (eps <= 0.000255)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.5e-6)
		tmp = sin(eps);
	elseif (eps <= 0.000255)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.5e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.000255], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.49999999999999995e-6 or 2.55e-4 < eps

   1. Initial program 59.3%

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

   2. Taylor expanded in x around 0 59.5%

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

   if -3.49999999999999995e-6 < eps < 2.55e-4

   1. Initial program 31.2%

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

   2. Taylor expanded in eps around 0 99.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000255:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9: 54.4% 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 46.1%

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

2. Taylor expanded in x around 0 56.8%

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

3. Final simplification 56.8%

   \[\leadsto \sin \varepsilon \]

Alternative 10: 28.7% 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 46.1%

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

2. Taylor expanded in eps around 0 48.7%

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

3. Taylor expanded in x around 0 27.2%

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

4. Final simplification 27.2%

   \[\leadsto \varepsilon \]

Developer target: 75.9% 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 2023240 
(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)))