2sin (example 3.3)

Percentage Accurate: 43.2% → 99.5%

Time: 13.5s

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: 43.2% 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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. add-exp-log 20.6%

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

3. Applied egg-rr 20.6%

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

   1. rem-exp-log 40.5%

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

   2. +-commutative 40.5%

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

   3. sin-sum 65.7%

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

   4. *-commutative 65.7%

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

   5. associate--l+ 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in x around inf 99.3%

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

   1. sub-neg 99.3%

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

   2. neg-mul-1 99.3%

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

   3. distribute-rgt-in 99.3%

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

   4. metadata-eval 99.3%

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

   5. sub-neg 99.3%

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

   6. *-commutative 99.3%

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

   7. sub-neg 99.3%

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

   8. metadata-eval 99.3%

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

8. Simplified 99.3%

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

   1. +-commutative 99.3%

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

   2. fma-def 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.05:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(\cos x \cdot 2\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (<= t_0 -0.05)
     t_0
     (if (<= t_0 0.0) (* (* (cos x) 2.0) (sin (* eps 0.5))) (sin eps)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (cos(x) * 2.0) * sin((eps * 0.5));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if (t_0 <= (-0.05d0)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (cos(x) * 2.0d0) * sin((eps * 0.5d0))
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (Math.cos(x) * 2.0) * Math.sin((eps * 0.5));
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if t_0 <= -0.05:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (math.cos(x) * 2.0) * math.sin((eps * 0.5))
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(cos(x) * 2.0) * sin(Float64(eps * 0.5)));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (cos(x) * 2.0) * sin((eps * 0.5));
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[Cos[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.050000000000000003

   1. Initial program 67.4%

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

   if -0.050000000000000003 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 0.0

   1. Initial program 18.1%

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

      1. diff-sin 18.1%

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

      2. div-inv 18.1%

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

      3. associate--l+ 18.1%

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

      4. metadata-eval 18.1%

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

      5. div-inv 18.1%

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

      6. +-commutative 18.1%

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

      7. associate-+l+ 18.1%

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

      8. metadata-eval 18.1%

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

   3. Applied egg-rr 18.1%

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

      1. associate-*r* 18.1%

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

      2. *-commutative 18.1%

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

      3. *-commutative 18.1%

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

      4. +-commutative 18.1%

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

      5. count-2 18.1%

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

      6. fma-def 18.1%

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

      7. associate-+r- 18.1%

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

      8. +-commutative 18.1%

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

      9. associate--l+ 81.2%

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

      10. +-inverses 81.2%

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

   5. Simplified 81.2%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right)} \]
   6. Step-by-step derivation

      1. add-cbrt-cube 80.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}} \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right) \]

      2. pow3 80.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}^{3}}} \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right) \]

   7. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}^{3}}} \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right) \]

   8. Taylor expanded in eps around 0 81.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\cos x}^{3}}} \cdot \left(2 \cdot \sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right)\right) \]

   9. Taylor expanded in x around inf 81.3%

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

       1. associate-*r* 81.3%

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

       2. *-commutative 81.3%

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

   11. Simplified 81.3%

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

   if 0.0 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 69.2%

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

   2. Taylor expanded in x around 0 69.5%

      \[\leadsto \color{blue}{\sin \varepsilon} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.05:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\sin \left(\varepsilon + x\right) - \sin x \leq 0:\\ \;\;\;\;\left(\cos x \cdot 2\right) \cdot \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 3: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.05:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (<= t_0 -0.05) t_0 (if (<= t_0 0.0) (* eps (cos x)) (sin eps)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * cos(x);
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if (t_0 <= (-0.05d0)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = eps * cos(x)
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if t_0 <= -0.05:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], t$95$0, If[LessEqual[t$95$0, 0.0], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.050000000000000003

   1. Initial program 67.4%

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

   if -0.050000000000000003 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 0.0

   1. Initial program 18.1%

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

   2. Taylor expanded in eps around 0 80.0%

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

   if 0.0 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 69.2%

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

   2. Taylor expanded in x around 0 69.5%

      \[\leadsto \color{blue}{\sin \varepsilon} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.05:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\sin \left(\varepsilon + x\right) - \sin x \leq 0:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. add-exp-log 20.6%

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

3. Applied egg-rr 20.6%

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

   1. rem-exp-log 40.5%

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

   2. +-commutative 40.5%

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

   3. sin-sum 65.7%

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

   4. *-commutative 65.7%

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

   5. associate--l+ 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in x around inf 99.3%

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

   1. sub-neg 99.3%

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

   2. neg-mul-1 99.3%

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

   3. distribute-rgt-in 99.3%

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

   4. metadata-eval 99.3%

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

   5. sub-neg 99.3%

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

   6. *-commutative 99.3%

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

   7. sub-neg 99.3%

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

   8. metadata-eval 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 5: 78.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. add-exp-log 20.6%

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

3. Applied egg-rr 20.6%

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

   1. rem-exp-log 40.5%

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

   2. +-commutative 40.5%

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

   3. sin-sum 65.7%

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

   4. *-commutative 65.7%

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

   5. associate--l+ 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in eps around 0 75.5%

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

7. Final simplification 75.5%

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

Alternative 6: 76.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. diff-sin 40.2%

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

   2. div-inv 40.2%

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

   3. associate--l+ 40.2%

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

   4. metadata-eval 40.2%

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

   5. div-inv 40.2%

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

   6. +-commutative 40.2%

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

   7. associate-+l+ 40.4%

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

   8. metadata-eval 40.4%

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

3. Applied egg-rr 40.4%

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

   1. associate-*r* 40.4%

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

   2. *-commutative 40.4%

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

   3. *-commutative 40.4%

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

   4. +-commutative 40.4%

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

   5. count-2 40.4%

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

   6. fma-def 40.4%

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

   7. associate-+r- 40.4%

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

   8. +-commutative 40.4%

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

   9. associate--l+ 75.3%

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

   10. +-inverses 75.3%

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

5. Simplified 75.3%

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

6. Taylor expanded in x around inf 75.3%

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

   1. *-commutative 75.3%

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

   2. associate-*r* 75.3%

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

   3. *-commutative 75.3%

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

8. Simplified 75.3%

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

9. Final simplification 75.3%

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

Alternative 7: 76.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. add-exp-log 20.6%

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

3. Applied egg-rr 20.6%

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

   1. rem-exp-log 40.5%

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

   2. +-commutative 40.5%

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

   3. sin-sum 65.7%

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

   4. *-commutative 65.7%

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

   5. associate--l+ 99.3%

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

5. Applied egg-rr 99.3%

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

   1. associate-+r- 65.7%

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

   2. *-commutative 65.7%

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

   3. sin-sum 40.5%

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

   4. +-commutative 40.5%

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

   5. diff-sin 40.2%

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

   6. *-commutative 40.2%

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

   7. div-inv 40.2%

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

   8. +-commutative 40.2%

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

   9. associate--l+ 75.3%

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

   10. metadata-eval 75.3%

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

   11. div-inv 75.3%

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

   12. +-commutative 75.3%

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

   13. metadata-eval 75.3%

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

7. Applied egg-rr 75.3%

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

8. Final simplification 75.3%

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

Alternative 8: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004 \lor \neg \left(\varepsilon \leq 0.000105\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0004) (not (<= eps 0.000105))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0004) || !(eps <= 0.000105)) {
		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 <= (-0.0004d0)) .or. (.not. (eps <= 0.000105d0))) 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 <= -0.0004) || !(eps <= 0.000105)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0004) or not (eps <= 0.000105):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0004) || !(eps <= 0.000105))
		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 <= -0.0004) || ~((eps <= 0.000105)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0004], N[Not[LessEqual[eps, 0.000105]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.00000000000000019e-4 or 1.05e-4 < eps

   1. Initial program 51.5%

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

   2. Taylor expanded in x around 0 51.9%

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

   if -4.00000000000000019e-4 < eps < 1.05e-4

   1. Initial program 29.2%

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

   2. Taylor expanded in eps around 0 98.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0004 \lor \neg \left(\varepsilon \leq 0.000105\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 56.1% 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 40.5%

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

2. Taylor expanded in x around 0 52.7%

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

3. Final simplification 52.7%

   \[\leadsto \sin \varepsilon \]

Alternative 10: 29.6% 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 40.5%

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

2. Taylor expanded in eps around 0 50.2%

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

3. Taylor expanded in x around 0 28.1%

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

4. Final simplification 28.1%

   \[\leadsto \varepsilon \]

Developer target: 99.5% 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 2023319 
(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)))