2sin (example 3.3)

Percentage Accurate: 42.1% → 99.4%

Time: 13.3s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.1% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) + (sin(x) * ((sin(eps) ** 2.0d0) / ((-1.0d0) - cos(eps))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.7%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

3. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. sub-neg 63.5%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. flip-+ 99.3%

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

   2. frac-2neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. sub-1-cos 99.5%

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

   5. pow2 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. remove-double-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-neg-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. sub-neg 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

3. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. sub-neg 63.5%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. fma-def 99.4%

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

   2. *-commutative 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

3. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. sub-neg 63.5%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-def 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

3. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. sub-neg 63.5%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 77.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

3. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. sub-neg 63.5%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around 0 76.1%

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

7. Final simplification 76.1%

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

Alternative 6: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.043:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -0.5 + \varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.043)
   (sin eps)
   (if (<= eps 2e-30)
     (+ (* (* (sin x) (* eps eps)) -0.5) (* eps (cos x)))
     (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.043) {
		tmp = sin(eps);
	} else if (eps <= 2e-30) {
		tmp = ((sin(x) * (eps * eps)) * -0.5) + (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) :: tmp
    if (eps <= (-0.043d0)) then
        tmp = sin(eps)
    else if (eps <= 2d-30) then
        tmp = ((sin(x) * (eps * eps)) * (-0.5d0)) + (eps * cos(x))
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.043) {
		tmp = Math.sin(eps);
	} else if (eps <= 2e-30) {
		tmp = ((Math.sin(x) * (eps * eps)) * -0.5) + (eps * Math.cos(x));
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.043:
		tmp = math.sin(eps)
	elif eps <= 2e-30:
		tmp = ((math.sin(x) * (eps * eps)) * -0.5) + (eps * math.cos(x))
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.043)
		tmp = sin(eps);
	elseif (eps <= 2e-30)
		tmp = Float64(Float64(Float64(sin(x) * Float64(eps * eps)) * -0.5) + Float64(eps * cos(x)));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.043)
		tmp = sin(eps);
	elseif (eps <= 2e-30)
		tmp = ((sin(x) * (eps * eps)) * -0.5) + (eps * cos(x));
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.043], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 2e-30], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.042999999999999997 or 2e-30 < eps

   1. Initial program 51.0%

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

   2. Taylor expanded in x around 0 52.6%

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

   if -0.042999999999999997 < eps < 2e-30

   1. Initial program 23.4%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-def 99.3%

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

      3. unpow2 99.3%

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

      4. associate-*l* 99.3%

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

      5. *-commutative 99.3%

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

   4. Simplified 99.3%

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

      1. fma-udef 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r* 99.3%

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

      4. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.043:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -0.5 + \varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

   1. diff-sin 37.6%

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

   2. div-inv 37.6%

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

   3. metadata-eval 37.6%

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

   4. div-inv 37.6%

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

   5. +-commutative 37.6%

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

   6. metadata-eval 37.6%

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

3. Applied egg-rr 37.6%

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

   1. *-commutative 37.6%

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

   2. +-commutative 37.6%

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

   3. associate--l+ 74.5%

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

   4. +-inverses 74.5%

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

   5. distribute-lft-in 74.5%

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

   6. metadata-eval 74.5%

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

   7. *-commutative 74.5%

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

   8. associate-+r+ 74.5%

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

   9. +-commutative 74.5%

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

5. Simplified 74.5%

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

6. Final simplification 74.5%

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

Alternative 8: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

   1. sin-sum 63.5%

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

   2. associate--l+ 63.5%

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

3. Applied egg-rr 63.5%

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

   1. +-commutative 63.5%

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

   2. sub-neg 63.5%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-def 99.4%

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

7. Applied egg-rr 99.4%

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

   1. fma-udef 99.4%

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

   2. *-commutative 99.4%

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

   3. +-commutative 99.4%

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

   4. distribute-rgt-in 99.4%

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

   5. neg-mul-1 99.4%

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

   6. associate-+r+ 63.5%

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

   7. sin-sum 37.7%

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

   8. +-commutative 37.7%

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

   9. add-cube-cbrt 36.7%

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

   10. fma-def 36.7%

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

   11. fma-neg 36.7%

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

   12. add-cube-cbrt 37.7%

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

   13. diff-sin 37.6%

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

9. Applied egg-rr 37.6%

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

    1. associate-*r* 37.6%

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

    2. *-commutative 37.6%

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

    3. +-commutative 37.6%

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

    4. +-commutative 37.6%

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

    5. associate--l+ 74.5%

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

    6. +-inverses 74.5%

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

11. Simplified 74.5%

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

12. Final simplification 74.5%

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

Alternative 9: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.043:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.043) (sin eps) (if (<= eps 2e-30) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.043) {
		tmp = sin(eps);
	} else if (eps <= 2e-30) {
		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) :: tmp
    if (eps <= (-0.043d0)) then
        tmp = sin(eps)
    else if (eps <= 2d-30) 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 tmp;
	if (eps <= -0.043) {
		tmp = Math.sin(eps);
	} else if (eps <= 2e-30) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.043:
		tmp = math.sin(eps)
	elif eps <= 2e-30:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.043)
		tmp = sin(eps);
	elseif (eps <= 2e-30)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.043)
		tmp = sin(eps);
	elseif (eps <= 2e-30)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.043], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 2e-30], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.042999999999999997 or 2e-30 < eps

   1. Initial program 51.0%

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

   2. Taylor expanded in x around 0 52.6%

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

   if -0.042999999999999997 < eps < 2e-30

   1. Initial program 23.4%

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

   2. Taylor expanded in eps around 0 99.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.043:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 10: 54.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.7%

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

2. Taylor expanded in x around 0 51.5%

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

3. Final simplification 51.5%

   \[\leadsto \sin \varepsilon \]

Alternative 11: 4.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 37.7%

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

   1. add-cube-cbrt 36.7%

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

   2. pow3 36.6%

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

3. Applied egg-rr 36.6%

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

4. Taylor expanded in eps around 0 4.2%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \sin x - \sin x} \]
5. Step-by-step derivation

   1. pow-base-1 4.2%

      \[\leadsto \color{blue}{1} \cdot \sin x - \sin x \]

   2. *-lft-identity 4.2%

      \[\leadsto \color{blue}{\sin x} - \sin x \]

   3. +-inverses 4.2%

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

6. Simplified 4.2%

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

7. Final simplification 4.2%

   \[\leadsto 0 \]

Alternative 12: 29.0% 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 37.7%

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

2. Taylor expanded in eps around 0 50.8%

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

3. Taylor expanded in x around 0 27.3%

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

4. Final simplification 27.3%

   \[\leadsto \varepsilon \]

Developer target: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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