2sin (example 3.3)

Percentage Accurate: 41.9% → 99.6%

Time: 13.8s

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: 41.9% 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: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[Sin[N[(eps * 0.5), $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. 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. +-commutative 69.1%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. flip-- 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. 1-sub-cos 99.6%

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

   6. pow2 99.6%

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

7. Applied egg-rr 99.6%

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

   1. flip-+ 51.5%

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

   2. metadata-eval 51.5%

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

   3. 1-sub-cos 72.8%

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

   4. unpow2 72.8%

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

   5. div-inv 72.8%

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

9. Applied egg-rr 72.8%

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

    1. associate-*r/ 72.8%

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

    2. *-rgt-identity 72.8%

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

    3. unpow2 72.8%

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

    4. associate-/l* 99.2%

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

    5. hang-p0-tan 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in eps around inf 99.6%

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

    1. associate-/l* 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (sin(eps) * cos(x)) - ((sin(x) * pow(sin(eps), 2.0)) / (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) * (sin(eps) ** 2.0d0)) / (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) * Math.pow(Math.sin(eps), 2.0)) / (1.0 + Math.cos(eps)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] * N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision]), $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. +-commutative 69.1%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. flip-- 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. 1-sub-cos 99.6%

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

   6. pow2 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $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. +-commutative 69.1%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

   1. fma-neg 99.3%

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

   2. distribute-rgt-neg-in 99.3%

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

   3. flip-- 99.2%

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

   4. distribute-neg-frac 99.2%

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

   5. metadata-eval 99.2%

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

   6. 1-sub-cos 99.6%

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

   7. sub-1-cos 99.2%

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

   8. metadata-eval 99.2%

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

   9. +-commutative 99.2%

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

   10. metadata-eval 99.2%

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

   11. sub-neg 99.2%

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

   12. flip-+ 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $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. +-commutative 69.1%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in eps around inf 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. cancel-sign-sub-inv 99.3%

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

   4. neg-sub0 99.3%

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

   5. associate--r- 99.3%

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

   6. metadata-eval 99.3%

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

   7. +-commutative 99.3%

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

   8. *-commutative 99.3%

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

   9. +-commutative 99.3%

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

   10. fma-def 99.3%

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

   11. *-commutative 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $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. +-commutative 69.1%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 6: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.2%

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

   1. add-cbrt-cube 37.7%

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

   2. pow3 37.7%

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

3. Applied egg-rr 37.7%

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

   1. rem-cbrt-cube 44.2%

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

   2. diff-sin 43.7%

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

   3. +-commutative 43.7%

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

   4. +-commutative 43.7%

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

5. Applied egg-rr 43.7%

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

   1. associate-*r* 43.7%

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

   2. *-commutative 43.7%

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

   3. associate-+l+ 43.8%

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

   4. count-2 43.8%

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

   5. associate--l+ 75.2%

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

   6. +-inverses 75.2%

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

7. Simplified 75.2%

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

8. Final simplification 75.2%

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

Alternative 7: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

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)) - 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 <= (-3.5d-6)) then
        tmp = sin(eps)
    else if (eps <= 6d-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 <= -3.5e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 6e-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 <= -3.5e-6:
		tmp = math.sin(eps)
	elif eps <= 6e-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 <= -3.5e-6)
		tmp = sin(eps);
	elseif (eps <= 6e-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 <= -3.5e-6)
		tmp = sin(eps);
	elseif (eps <= 6e-5)
		tmp = eps * cos(x);
	else
		tmp = sin((eps + x)) - sin(x);
	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[Sin[x], $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%

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

4. Final simplification 75.1%

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

Alternative 8: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -7.2e-5) (not (<= eps 6.6e-6))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.2e-5) || !(eps <= 6.6e-6)) {
		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 <= (-7.2d-5)) .or. (.not. (eps <= 6.6d-6))) 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 <= -7.2e-5) || !(eps <= 6.6e-6)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -7.2e-5) or not (eps <= 6.6e-6):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -7.2e-5) || !(eps <= 6.6e-6))
		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 <= -7.2e-5) || ~((eps <= 6.6e-6)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -7.2e-5], N[Not[LessEqual[eps, 6.6e-6]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.20000000000000018e-5 or 6.60000000000000034e-6 < 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 -7.20000000000000018e-5 < eps < 6.60000000000000034e-6

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 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 10: 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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

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

  :herbie-target
  (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x)))

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