2sin (example 3.3)

Percentage Accurate: 41.8% → 99.4%

Time: 11.6s

Alternatives: 10

Speedup: 2.0×

• could not determine a ground truth

  1. x = 7.754657567888337e+285
  2. eps = 5.292099588805758e+167

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.1%

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

3. Applied egg-rr 65.1%

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

   1. +-commutative 65.1%

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

   2. associate-+l- 99.6%

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

   3. *-commutative 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. distribute-lft-out-- 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in eps around inf 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-rgt-neg-out 99.7%

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

   4. +-commutative 99.7%

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

   5. fma-def 99.7%

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

   6. neg-sub0 99.7%

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

   7. associate--r- 99.7%

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

   8. metadata-eval 99.7%

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

   9. +-commutative 99.7%

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

   10. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.9%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.1%

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

3. Applied egg-rr 65.1%

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

   1. +-commutative 65.1%

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

   2. associate-+l- 99.6%

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

   3. *-commutative 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. distribute-lft-out-- 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon + t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin x) (+ (cos eps) -1.0))))
   (if (or (<= eps -14200000.0) (not (<= eps 1.55e-10)))
     (+ (sin eps) t_0)
     (+ (* eps (cos x)) t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(x) * (cos(eps) + -1.0);
	double tmp;
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10)) {
		tmp = sin(eps) + t_0;
	} else {
		tmp = (eps * cos(x)) + t_0;
	}
	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(x) * (cos(eps) + (-1.0d0))
    if ((eps <= (-14200000.0d0)) .or. (.not. (eps <= 1.55d-10))) then
        tmp = sin(eps) + t_0
    else
        tmp = (eps * cos(x)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) * (Math.cos(eps) + -1.0);
	double tmp;
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10)) {
		tmp = Math.sin(eps) + t_0;
	} else {
		tmp = (eps * Math.cos(x)) + t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(x) * (math.cos(eps) + -1.0)
	tmp = 0
	if (eps <= -14200000.0) or not (eps <= 1.55e-10):
		tmp = math.sin(eps) + t_0
	else:
		tmp = (eps * math.cos(x)) + t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(x) * Float64(cos(eps) + -1.0))
	tmp = 0.0
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10))
		tmp = Float64(sin(eps) + t_0);
	else
		tmp = Float64(Float64(eps * cos(x)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) * (cos(eps) + -1.0);
	tmp = 0.0;
	if ((eps <= -14200000.0) || ~((eps <= 1.55e-10)))
		tmp = sin(eps) + t_0;
	else
		tmp = (eps * cos(x)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -14200000.0], N[Not[LessEqual[eps, 1.55e-10]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.42e7 or 1.55000000000000008e-10 < eps

   1. Initial program 54.0%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.4%

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

   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. 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-- 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 57.3%

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

   if -1.42e7 < eps < 1.55000000000000008e-10

   1. Initial program 30.7%

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

      1. sin-sum 33.5%

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

      2. associate--l+ 33.5%

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

   3. Applied egg-rr 33.5%

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

      1. +-commutative 33.5%

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

      2. associate-+l- 99.7%

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

      3. *-commutative 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. distribute-lft-out-- 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in eps around 0 97.6%

      \[\leadsto \color{blue}{\varepsilon \cdot \cos x} - \sin x \cdot \left(1 - \cos \varepsilon\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \]

Alternative 4: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -14200000.0) (not (<= eps 1.55e-10)))
   (+ (sin eps) (* (sin x) (+ (cos eps) -1.0)))
   (* (cos x) (* 2.0 (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10)) {
		tmp = sin(eps) + (sin(x) * (cos(eps) + -1.0));
	} else {
		tmp = cos(x) * (2.0 * sin((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 <= (-14200000.0d0)) .or. (.not. (eps <= 1.55d-10))) then
        tmp = sin(eps) + (sin(x) * (cos(eps) + (-1.0d0)))
    else
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10)) {
		tmp = Math.sin(eps) + (Math.sin(x) * (Math.cos(eps) + -1.0));
	} else {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -14200000.0) or not (eps <= 1.55e-10):
		tmp = math.sin(eps) + (math.sin(x) * (math.cos(eps) + -1.0))
	else:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10))
		tmp = Float64(sin(eps) + Float64(sin(x) * Float64(cos(eps) + -1.0)));
	else
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -14200000.0) || ~((eps <= 1.55e-10)))
		tmp = sin(eps) + (sin(x) * (cos(eps) + -1.0));
	else
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -14200000.0], N[Not[LessEqual[eps, 1.55e-10]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.42e7 or 1.55000000000000008e-10 < eps

   1. Initial program 54.0%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.4%

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

   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. 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-- 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 57.3%

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

   if -1.42e7 < eps < 1.55000000000000008e-10

   1. Initial program 30.7%

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

      1. diff-sin 30.7%

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

      2. div-inv 30.7%

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

      3. associate--l+ 30.7%

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

      4. metadata-eval 30.7%

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

      5. div-inv 30.7%

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

      6. +-commutative 30.7%

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

      7. associate-+l+ 30.7%

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

      8. metadata-eval 30.7%

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

   3. Applied egg-rr 30.7%

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

      1. associate-*r* 30.7%

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

      2. *-commutative 30.7%

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

      3. *-commutative 30.7%

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

      4. +-commutative 30.7%

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

      5. count-2 30.7%

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

      6. fma-def 30.7%

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

      7. sub-neg 30.7%

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

      8. mul-1-neg 30.7%

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

      9. +-commutative 30.7%

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

      10. associate-+r+ 97.6%

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

      11. mul-1-neg 97.6%

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

      12. sub-neg 97.6%

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

      13. +-inverses 97.6%

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

      14. remove-double-neg 97.6%

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

      15. mul-1-neg 97.6%

          \[\leadsto \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 97.6%

          \[\leadsto \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 97.6%

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

      18. mul-1-neg 97.6%

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

      19. remove-double-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in eps around 0 97.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-8}:\\ \;\;\;\;\sin \varepsilon + \sin x \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon + x \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)))
   (if (<= eps -3e-8)
     (+ (sin eps) (* (sin x) t_0))
     (+ (* (cos x) (sin eps)) (* x t_0)))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double tmp;
	if (eps <= -3e-8) {
		tmp = sin(eps) + (sin(x) * t_0);
	} else {
		tmp = (cos(x) * sin(eps)) + (x * t_0);
	}
	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 = cos(eps) + (-1.0d0)
    if (eps <= (-3d-8)) then
        tmp = sin(eps) + (sin(x) * t_0)
    else
        tmp = (cos(x) * sin(eps)) + (x * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) + -1.0;
	double tmp;
	if (eps <= -3e-8) {
		tmp = Math.sin(eps) + (Math.sin(x) * t_0);
	} else {
		tmp = (Math.cos(x) * Math.sin(eps)) + (x * t_0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	tmp = 0
	if eps <= -3e-8:
		tmp = math.sin(eps) + (math.sin(x) * t_0)
	else:
		tmp = (math.cos(x) * math.sin(eps)) + (x * t_0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	tmp = 0.0
	if (eps <= -3e-8)
		tmp = Float64(sin(eps) + Float64(sin(x) * t_0));
	else
		tmp = Float64(Float64(cos(x) * sin(eps)) + Float64(x * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	tmp = 0.0;
	if (eps <= -3e-8)
		tmp = sin(eps) + (sin(x) * t_0);
	else
		tmp = (cos(x) * sin(eps)) + (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[eps, -3e-8], N[(N[Sin[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.99999999999999973e-8

   1. Initial program 56.4%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   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+ 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. distribute-lft-out-- 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 59.6%

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

   if -2.99999999999999973e-8 < eps

   1. Initial program 35.9%

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

      1. sin-sum 51.0%

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

      2. associate--l+ 51.0%

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

   3. Applied egg-rr 51.0%

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

      1. +-commutative 51.0%

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

      2. associate-+l- 99.6%

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

      3. *-commutative 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. distribute-lft-out-- 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 84.2%

      \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{x \cdot \left(1 - \cos \varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-8}:\\ \;\;\;\;\sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \]

Alternative 6: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sin \varepsilon + \sin x \cdot 0 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return (Math.cos(x) * Math.sin(eps)) + (Math.sin(x) * 0.0);
}

Python

def code(x, eps):
	return (math.cos(x) * math.sin(eps)) + (math.sin(x) * 0.0)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (cos(x) * sin(eps)) + (sin(x) * 0.0);
end

Wolfram

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

Derivation

1. Initial program 41.9%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.1%

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

3. Applied egg-rr 65.1%

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

   1. +-commutative 65.1%

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

   2. associate-+l- 99.6%

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

   3. *-commutative 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. distribute-lft-out-- 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in eps around 0 78.3%

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

7. Final simplification 78.3%

   \[\leadsto \cos x \cdot \sin \varepsilon + \sin x \cdot 0 \]

Alternative 7: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -14200000.0) (not (<= eps 1.55e-10)))
   (sin eps)
   (* (cos x) (* 2.0 (sin (* eps 0.5))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10)) {
		tmp = sin(eps);
	} else {
		tmp = cos(x) * (2.0 * sin((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 <= (-14200000.0d0)) .or. (.not. (eps <= 1.55d-10))) then
        tmp = sin(eps)
    else
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10)) {
		tmp = Math.sin(eps);
	} else {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -14200000.0) or not (eps <= 1.55e-10):
		tmp = math.sin(eps)
	else:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10))
		tmp = sin(eps);
	else
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -14200000.0) || ~((eps <= 1.55e-10)))
		tmp = sin(eps);
	else
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -14200000.0], N[Not[LessEqual[eps, 1.55e-10]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.42e7 or 1.55000000000000008e-10 < eps

   1. Initial program 54.0%

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

   2. Taylor expanded in x around 0 55.5%

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

   if -1.42e7 < eps < 1.55000000000000008e-10

   1. Initial program 30.7%

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

      1. diff-sin 30.7%

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

      2. div-inv 30.7%

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

      3. associate--l+ 30.7%

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

      4. metadata-eval 30.7%

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

      5. div-inv 30.7%

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

      6. +-commutative 30.7%

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

      7. associate-+l+ 30.7%

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

      8. metadata-eval 30.7%

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

   3. Applied egg-rr 30.7%

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

      1. associate-*r* 30.7%

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

      2. *-commutative 30.7%

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

      3. *-commutative 30.7%

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

      4. +-commutative 30.7%

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

      5. count-2 30.7%

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

      6. fma-def 30.7%

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

      7. sub-neg 30.7%

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

      8. mul-1-neg 30.7%

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

      9. +-commutative 30.7%

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

      10. associate-+r+ 97.6%

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

      11. mul-1-neg 97.6%

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

      12. sub-neg 97.6%

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

      13. +-inverses 97.6%

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

      14. remove-double-neg 97.6%

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

      15. mul-1-neg 97.6%

          \[\leadsto \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 97.6%

          \[\leadsto \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 97.6%

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

      18. mul-1-neg 97.6%

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

      19. remove-double-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in eps around 0 97.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -14200000.0) (not (<= eps 1.55e-10)))
   (sin eps)
   (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10)) {
		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 <= (-14200000.0d0)) .or. (.not. (eps <= 1.55d-10))) 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 <= -14200000.0) || !(eps <= 1.55e-10)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -14200000.0) or not (eps <= 1.55e-10):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -14200000.0) || !(eps <= 1.55e-10))
		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 <= -14200000.0) || ~((eps <= 1.55e-10)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -14200000.0], N[Not[LessEqual[eps, 1.55e-10]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.42e7 or 1.55000000000000008e-10 < eps

   1. Initial program 54.0%

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

   2. Taylor expanded in x around 0 55.5%

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

   if -1.42e7 < eps < 1.55000000000000008e-10

   1. Initial program 30.7%

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

   2. Taylor expanded in eps around 0 97.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14200000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-10}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 54.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (sin eps))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return math.sin(eps)

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 41.9%

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

2. Taylor expanded in x around 0 54.7%

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

3. Final simplification 54.7%

   \[\leadsto \sin \varepsilon \]

Alternative 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 41.9%

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

2. Taylor expanded in eps around 0 52.7%

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

3. Taylor expanded in x around 0 30.3%

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

4. Final simplification 30.3%

   \[\leadsto \varepsilon \]

Developer target: 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]

Reproduce

herbie shell --seed 2023313 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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