2sin (example 3.3)

Percentage Accurate: 42.9% → 99.3%

Time: 11.8s

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: 42.9% 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: 99.3% 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 40.2%

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

   1. sin-sum 64.6%

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

   2. associate--l+ 64.6%

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

3. Applied egg-rr 64.6%

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

   1. +-commutative 64.6%

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

   2. sub-neg 64.6%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.3%

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

   8. +-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in eps around inf 99.3%

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

   1. fma-def 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

   1. sin-sum 64.6%

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

   2. associate--l+ 64.6%

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

3. Applied egg-rr 64.6%

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

   1. +-commutative 64.6%

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

   2. sub-neg 64.6%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.3%

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

   8. +-commutative 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(\varepsilon - x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00014) (not (<= eps 5.5e-5)))
   (- (sin (- eps x)) (sin x))
   (+ (* -0.5 (* (sin x) (* eps eps))) (* (cos x) eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00014) || !(eps <= 5.5e-5)) {
		tmp = sin((eps - x)) - sin(x);
	} else {
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (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 <= (-0.00014d0)) .or. (.not. (eps <= 5.5d-5))) then
        tmp = sin((eps - x)) - sin(x)
    else
        tmp = ((-0.5d0) * (sin(x) * (eps * eps))) + (cos(x) * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00014) || !(eps <= 5.5e-5)) {
		tmp = Math.sin((eps - x)) - Math.sin(x);
	} else {
		tmp = (-0.5 * (Math.sin(x) * (eps * eps))) + (Math.cos(x) * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00014) or not (eps <= 5.5e-5):
		tmp = math.sin((eps - x)) - math.sin(x)
	else:
		tmp = (-0.5 * (math.sin(x) * (eps * eps))) + (math.cos(x) * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00014) || !(eps <= 5.5e-5))
		tmp = Float64(sin(Float64(eps - x)) - sin(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * Float64(eps * eps))) + Float64(cos(x) * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00014) || ~((eps <= 5.5e-5)))
		tmp = sin((eps - x)) - sin(x);
	else
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (cos(x) * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00014], N[Not[LessEqual[eps, 5.5e-5]], $MachinePrecision]], N[(N[Sin[N[(eps - x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.3999999999999999e-4 or 5.5000000000000002e-5 < eps

   1. Initial program 52.0%

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

      1. sin-sum 99.5%

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

      2. associate--l+ 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. sub-neg 99.4%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. neg-mul-1 99.5%

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

      6. *-commutative 99.5%

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

      7. distribute-rgt-out 99.4%

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

      8. +-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. add-sqr-sqrt 46.9%

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

      2. add-sqr-sqrt 46.9%

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

      3. sqr-neg 46.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 24.0%

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

      6. distribute-rgt-neg-in 24.0%

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

      7. add-sqr-sqrt 52.3%

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

      8. cancel-sign-sub-inv 52.3%

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

      9. distribute-rgt-in 52.3%

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

      10. neg-mul-1 52.3%

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

      11. associate--r+ 52.3%

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

      12. sin-diff 49.0%

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

      13. add-sqr-sqrt 22.1%

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

   7. Applied egg-rr 54.3%

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

   if -1.3999999999999999e-4 < eps < 5.5000000000000002e-5

   1. Initial program 28.3%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. unpow2 99.7%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

   4. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. *-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(\varepsilon - x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \cos x \cdot \varepsilon\\ \end{array} \]

Alternative 4: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

   1. diff-sin 39.6%

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

   2. div-inv 39.6%

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

   3. metadata-eval 39.6%

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

   4. div-inv 39.6%

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

   5. +-commutative 39.6%

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

   6. metadata-eval 39.6%

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

3. Applied egg-rr 39.6%

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

   1. *-commutative 39.6%

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

   2. +-commutative 39.6%

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

   3. associate--l+ 75.3%

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

   4. +-inverses 75.3%

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

   5. distribute-lft-in 75.3%

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

   6. metadata-eval 75.3%

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

   7. *-commutative 75.3%

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

   8. associate-+r+ 75.3%

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

   9. +-commutative 75.3%

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

5. Simplified 75.3%

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

6. Final simplification 75.3%

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

Alternative 5: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(\varepsilon - x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00015) (not (<= eps 5.5e-5)))
   (- (sin (- eps x)) (sin x))
   (* (cos x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00015) || !(eps <= 5.5e-5)) {
		tmp = sin((eps - x)) - sin(x);
	} 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 <= (-0.00015d0)) .or. (.not. (eps <= 5.5d-5))) then
        tmp = sin((eps - x)) - sin(x)
    else
        tmp = cos(x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00015) || !(eps <= 5.5e-5)) {
		tmp = Math.sin((eps - x)) - Math.sin(x);
	} else {
		tmp = Math.cos(x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00015) or not (eps <= 5.5e-5):
		tmp = math.sin((eps - x)) - math.sin(x)
	else:
		tmp = math.cos(x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00015) || !(eps <= 5.5e-5))
		tmp = Float64(sin(Float64(eps - x)) - sin(x));
	else
		tmp = Float64(cos(x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00015) || ~((eps <= 5.5e-5)))
		tmp = sin((eps - x)) - sin(x);
	else
		tmp = cos(x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00015], N[Not[LessEqual[eps, 5.5e-5]], $MachinePrecision]], N[(N[Sin[N[(eps - x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.49999999999999987e-4 or 5.5000000000000002e-5 < eps

   1. Initial program 52.0%

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

      1. sin-sum 99.5%

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

      2. associate--l+ 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. sub-neg 99.4%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. neg-mul-1 99.5%

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

      6. *-commutative 99.5%

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

      7. distribute-rgt-out 99.4%

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

      8. +-commutative 99.4%

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

   5. Simplified 99.4%

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

      1. add-sqr-sqrt 46.9%

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

      2. add-sqr-sqrt 46.9%

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

      3. sqr-neg 46.9%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 24.0%

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

      6. distribute-rgt-neg-in 24.0%

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

      7. add-sqr-sqrt 52.3%

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

      8. cancel-sign-sub-inv 52.3%

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

      9. distribute-rgt-in 52.3%

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

      10. neg-mul-1 52.3%

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

      11. associate--r+ 52.3%

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

      12. sin-diff 49.0%

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

      13. add-sqr-sqrt 22.1%

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

   7. Applied egg-rr 54.3%

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

   if -1.49999999999999987e-4 < eps < 5.5000000000000002e-5

   1. Initial program 28.3%

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

   2. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00015 \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(\varepsilon - x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]

Alternative 6: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0014:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3e-5)
   (sin eps)
   (if (<= eps 0.0014) (* (cos x) eps) (- (sin (+ x eps)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3e-5) {
		tmp = sin(eps);
	} else if (eps <= 0.0014) {
		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 <= (-3d-5)) then
        tmp = sin(eps)
    else if (eps <= 0.0014d0) 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 <= -3e-5) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.0014) {
		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 <= -3e-5:
		tmp = math.sin(eps)
	elif eps <= 0.0014:
		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 <= -3e-5)
		tmp = sin(eps);
	elseif (eps <= 0.0014)
		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 <= -3e-5)
		tmp = sin(eps);
	elseif (eps <= 0.0014)
		tmp = cos(x) * eps;
	else
		tmp = sin((x + eps)) - sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3e-5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.0014], 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 < -3.00000000000000008e-5

   1. Initial program 49.8%

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

   2. Taylor expanded in x around 0 50.7%

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

   if -3.00000000000000008e-5 < eps < 0.00139999999999999999

   1. Initial program 28.3%

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

   2. Taylor expanded in eps around 0 99.1%

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

   if 0.00139999999999999999 < eps

   1. Initial program 55.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0014:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \end{array} \]

Alternative 7: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.6e-5)
   (sin eps)
   (if (<= eps 2.95e-5) (* (cos x) eps) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.6e-5) {
		tmp = sin(eps);
	} else if (eps <= 2.95e-5) {
		tmp = cos(x) * eps;
	} 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 <= (-3.6d-5)) then
        tmp = sin(eps)
    else if (eps <= 2.95d-5) then
        tmp = cos(x) * eps
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3.6e-5) {
		tmp = Math.sin(eps);
	} else if (eps <= 2.95e-5) {
		tmp = Math.cos(x) * eps;
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.6e-5:
		tmp = math.sin(eps)
	elif eps <= 2.95e-5:
		tmp = math.cos(x) * eps
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.6e-5)
		tmp = sin(eps);
	elseif (eps <= 2.95e-5)
		tmp = Float64(cos(x) * eps);
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.6e-5)
		tmp = sin(eps);
	elseif (eps <= 2.95e-5)
		tmp = cos(x) * eps;
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.6e-5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 2.95e-5], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.60000000000000009e-5 or 2.9499999999999999e-5 < eps

   1. Initial program 52.0%

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

   2. Taylor expanded in x around 0 52.3%

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

   if -3.60000000000000009e-5 < eps < 2.9499999999999999e-5

   1. Initial program 28.3%

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

   2. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8: 55.6% 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 40.2%

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

2. Taylor expanded in x around 0 53.9%

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

3. Final simplification 53.9%

   \[\leadsto \sin \varepsilon \]

Alternative 9: 4.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 40.2%

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

   1. sub-neg 40.2%

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

   2. +-commutative 40.2%

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

   3. add-sqr-sqrt 17.4%

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

   4. distribute-rgt-neg-in 17.4%

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

   5. fma-def 17.1%

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

3. Applied egg-rr 17.1%

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

4. Taylor expanded in eps around 0 4.4%

   \[\leadsto \color{blue}{-1 \cdot \sin x + \sin x} \]
5. Step-by-step derivation

   1. distribute-lft1-in 4.4%

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

   2. metadata-eval 4.4%

      \[\leadsto \color{blue}{0} \cdot \sin x \]

   3. mul0-lft 4.4%

      \[\leadsto \color{blue}{0} \]

6. Simplified 4.4%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 4.4%

   \[\leadsto 0 \]

Alternative 10: 29.3% 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 40.2%

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

2. Taylor expanded in eps around 0 51.0%

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

3. Taylor expanded in x around 0 29.4%

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

4. Final simplification 29.4%

   \[\leadsto \varepsilon \]

Developer target: 76.4% 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 2023171 
(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)))