2sin (example 3.3)

Percentage Accurate: 42.5% → 99.4%

Time: 12.1s

Alternatives: 8

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: 42.5% 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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

   1. sin-sum 70.9%

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

   2. associate--l+ 70.8%

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

3. Applied egg-rr 70.8%

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

   1. +-commutative 70.8%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

   1. fma-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. sub-neg 99.5%

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

   4. +-commutative 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. remove-double-neg 99.5%

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

   7. metadata-eval 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

   1. sin-sum 70.9%

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

   2. associate--l+ 70.8%

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

3. Applied egg-rr 70.8%

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

   1. +-commutative 70.8%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000185:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right) \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -8.5)
   (sin eps)
   (if (<= eps 0.000185)
     (* (cos x) eps)
     (* 2.0 (* (sin (* 0.5 (+ x (+ x eps)))) (cos (* eps 0.5)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5) {
		tmp = sin(eps);
	} else if (eps <= 0.000185) {
		tmp = cos(x) * eps;
	} else {
		tmp = 2.0 * (sin((0.5 * (x + (x + eps)))) * cos((eps * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-8.5d0)) then
        tmp = sin(eps)
    else if (eps <= 0.000185d0) then
        tmp = cos(x) * eps
    else
        tmp = 2.0d0 * (sin((0.5d0 * (x + (x + eps)))) * cos((eps * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.000185) {
		tmp = Math.cos(x) * eps;
	} else {
		tmp = 2.0 * (Math.sin((0.5 * (x + (x + eps)))) * Math.cos((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -8.5:
		tmp = math.sin(eps)
	elif eps <= 0.000185:
		tmp = math.cos(x) * eps
	else:
		tmp = 2.0 * (math.sin((0.5 * (x + (x + eps)))) * math.cos((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -8.5)
		tmp = sin(eps);
	elseif (eps <= 0.000185)
		tmp = Float64(cos(x) * eps);
	else
		tmp = Float64(2.0 * Float64(sin(Float64(0.5 * Float64(x + Float64(x + eps)))) * cos(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -8.5)
		tmp = sin(eps);
	elseif (eps <= 0.000185)
		tmp = cos(x) * eps;
	else
		tmp = 2.0 * (sin((0.5 * (x + (x + eps)))) * cos((eps * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -8.5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.000185], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision], N[(2.0 * N[(N[Sin[N[(0.5 * N[(x + N[(x + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -8.5

   1. Initial program 49.3%

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

   2. Taylor expanded in x around 0 53.0%

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

   if -8.5 < eps < 1.85e-4

   1. Initial program 34.8%

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

   2. Taylor expanded in eps around 0 98.1%

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

   if 1.85e-4 < eps

   1. Initial program 48.4%

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

      1. sin-sum 99.4%

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

      2. associate--l+ 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. associate-+l- 99.4%

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

      3. *-commutative 99.4%

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

      4. *-rgt-identity 99.4%

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

      5. distribute-lft-out-- 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in eps around inf 99.4%

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

      1. fma-neg 99.5%

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

      2. distribute-rgt-neg-in 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. remove-double-neg 99.5%

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

      7. metadata-eval 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.5%

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

      2. neg-mul-1 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. fma-udef 99.4%

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

       2. associate-+r+ 99.4%

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

       3. *-commutative 99.4%

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

       4. sin-sum 48.4%

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

       5. +-commutative 48.4%

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

       6. add-sqr-sqrt 26.5%

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

       7. sqrt-unprod 47.7%

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

       8. sqr-neg 47.7%

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

       9. sqrt-unprod 21.3%

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

       10. add-sqr-sqrt 46.6%

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

       11. sum-sin 47.9%

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

       12. div-inv 47.9%

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

       13. associate-+l+ 47.9%

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

       14. metadata-eval 47.9%

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

       15. div-inv 47.9%

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

   12. Applied egg-rr 49.9%

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

       1. *-commutative 49.9%

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

       2. +-inverses 49.9%

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

       3. +-rgt-identity 49.9%

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

   14. Simplified 49.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000185:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(0.5 \cdot \left(x + \left(x + \varepsilon\right)\right)\right) \cdot \cos \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

   1. sin-sum 70.9%

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

   2. associate--l+ 70.8%

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

3. Applied egg-rr 70.8%

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

   1. +-commutative 70.8%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

   1. fma-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. sub-neg 99.5%

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

   4. +-commutative 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. remove-double-neg 99.5%

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

   7. metadata-eval 99.5%

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

8. Simplified 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. neg-mul-1 99.5%

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

10. Applied egg-rr 99.5%

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

    1. fma-udef 99.4%

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

    2. associate-+r+ 70.9%

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

    3. *-commutative 70.9%

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

    4. sin-sum 42.4%

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

    5. +-commutative 42.4%

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

    6. sub-neg 42.4%

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

    7. diff-sin 42.0%

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

    8. div-inv 42.0%

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

    9. +-commutative 42.0%

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

    10. associate--l+ 72.1%

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

    11. metadata-eval 72.1%

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

    12. div-inv 72.1%

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

    13. associate-+l+ 72.1%

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

    14. metadata-eval 72.1%

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

12. Applied egg-rr 72.1%

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

    1. *-commutative 72.1%

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

    2. associate-*l* 72.1%

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

    3. +-inverses 72.1%

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

    4. +-rgt-identity 72.1%

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

    5. *-commutative 72.1%

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

14. Simplified 72.1%

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

15. Final simplification 72.1%

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

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4000:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -8.5)
   (sin eps)
   (if (<= eps 4000.0) (* (cos x) eps) (- (sin (+ x eps)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5) {
		tmp = sin(eps);
	} else if (eps <= 4000.0) {
		tmp = cos(x) * eps;
	} else {
		tmp = sin((x + eps)) - 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 <= (-8.5d0)) then
        tmp = sin(eps)
    else if (eps <= 4000.0d0) then
        tmp = cos(x) * eps
    else
        tmp = sin((x + eps)) - sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5) {
		tmp = Math.sin(eps);
	} else if (eps <= 4000.0) {
		tmp = Math.cos(x) * eps;
	} else {
		tmp = Math.sin((x + eps)) - Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -8.5:
		tmp = math.sin(eps)
	elif eps <= 4000.0:
		tmp = math.cos(x) * eps
	else:
		tmp = math.sin((x + eps)) - math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -8.5)
		tmp = sin(eps);
	elseif (eps <= 4000.0)
		tmp = Float64(cos(x) * eps);
	else
		tmp = Float64(sin(Float64(x + eps)) - sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -8.5)
		tmp = sin(eps);
	elseif (eps <= 4000.0)
		tmp = cos(x) * eps;
	else
		tmp = sin((x + eps)) - sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -8.5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 4000.0], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision], N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -8.5

   1. Initial program 49.3%

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

   2. Taylor expanded in x around 0 53.0%

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

   if -8.5 < eps < 4e3

   1. Initial program 34.0%

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

   2. Taylor expanded in eps around 0 95.7%

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

   if 4e3 < eps

   1. Initial program 50.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4000:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \end{array} \]

Alternative 6: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \lor \neg \left(\varepsilon \leq 4000\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8.5) (not (<= eps 4000.0))) (sin eps) (* (cos x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.5) || !(eps <= 4000.0)) {
		tmp = sin(eps);
	} else {
		tmp = cos(x) * eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-8.5d0)) .or. (.not. (eps <= 4000.0d0))) then
        tmp = sin(eps)
    else
        tmp = cos(x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.5) || !(eps <= 4000.0)) {
		tmp = Math.sin(eps);
	} else {
		tmp = Math.cos(x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -8.5) or not (eps <= 4000.0):
		tmp = math.sin(eps)
	else:
		tmp = math.cos(x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8.5) || !(eps <= 4000.0))
		tmp = sin(eps);
	else
		tmp = Float64(cos(x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -8.5) || ~((eps <= 4000.0)))
		tmp = sin(eps);
	else
		tmp = cos(x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8.5], N[Not[LessEqual[eps, 4000.0]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.5 or 4e3 < eps

   1. Initial program 49.8%

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

   2. Taylor expanded in x around 0 51.5%

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

   if -8.5 < eps < 4e3

   1. Initial program 34.0%

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

   2. Taylor expanded in eps around 0 95.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \lor \neg \left(\varepsilon \leq 4000\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]

Alternative 7: 54.7% 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 42.4%

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

2. Taylor expanded in x around 0 54.7%

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

3. Final simplification 54.7%

   \[\leadsto \sin \varepsilon \]

Alternative 8: 28.8% 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 42.4%

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

2. Taylor expanded in eps around 0 46.8%

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

3. Taylor expanded in x around 0 29.2%

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

4. Final simplification 29.2%

   \[\leadsto \varepsilon \]

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 2023320 
(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)))