2sin (example 3.3)

Percentage Accurate: 42.2% → 99.4%

Time: 11.8s

Alternatives: 9

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: 42.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. sin-sum 67.3%

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

   2. associate--l+ 67.3%

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

3. Applied egg-rr 67.3%

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

   1. +-commutative 67.3%

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

   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-- 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in eps around inf 99.5%

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

   1. fma-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. sub-neg 99.5%

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

   4. distribute-neg-in 99.5%

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

   5. metadata-eval 99.5%

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

   6. remove-double-neg 99.5%

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

   7. +-commutative 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in x around inf 99.5%

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

    1. +-commutative 99.5%

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

    2. *-commutative 99.5%

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

    3. fma-def 99.6%

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

    4. sub-neg 99.6%

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

    5. metadata-eval 99.6%

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

    6. *-commutative 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. sin-sum 67.3%

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

   2. associate--l+ 67.3%

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

3. Applied egg-rr 67.3%

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

   1. +-commutative 67.3%

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

   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-- 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in eps around inf 99.5%

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

   1. fma-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. sub-neg 99.5%

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

   4. distribute-neg-in 99.5%

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

   5. metadata-eval 99.5%

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

   6. remove-double-neg 99.5%

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

   7. +-commutative 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. sin-sum 67.3%

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

   2. associate--l+ 67.3%

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

3. Applied egg-rr 67.3%

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

   1. +-commutative 67.3%

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

   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-- 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot 0\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. sin-sum 67.3%

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

   2. associate--l+ 67.3%

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

3. Applied egg-rr 67.3%

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

   1. +-commutative 67.3%

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

   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-- 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in eps around inf 99.5%

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

   1. fma-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. sub-neg 99.5%

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

   4. distribute-neg-in 99.5%

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

   5. metadata-eval 99.5%

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

   6. remove-double-neg 99.5%

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

   7. +-commutative 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in eps around 0 75.3%

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

10. Final simplification 75.3%

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

Alternative 5: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

   1. add-cbrt-cube 36.6%

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

   2. pow3 36.6%

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

3. Applied egg-rr 36.6%

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

   1. rem-cbrt-cube 41.8%

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

   2. diff-sin 41.2%

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

   3. div-inv 41.2%

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

   4. +-commutative 41.2%

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

   5. associate--l+ 74.0%

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

   6. metadata-eval 74.0%

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

   7. div-inv 74.0%

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

   8. +-commutative 74.0%

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

   9. associate-+l+ 73.9%

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

   10. metadata-eval 73.9%

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

5. Applied egg-rr 73.9%

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

   1. associate-*r* 73.9%

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

   2. *-commutative 73.9%

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

   3. *-commutative 73.9%

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

   4. count-2 73.9%

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

   5. *-commutative 73.9%

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

   6. +-inverses 73.9%

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

   7. +-rgt-identity 73.9%

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

7. Simplified 73.9%

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

8. Final simplification 73.9%

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

Alternative 6: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -520000000000:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -520000000000.0)
   (- (sin (+ eps x)) (sin x))
   (if (<= eps 6e-6) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -520000000000.0) {
		tmp = sin((eps + x)) - sin(x);
	} else if (eps <= 6e-6) {
		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 <= (-520000000000.0d0)) then
        tmp = sin((eps + x)) - sin(x)
    else if (eps <= 6d-6) 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 <= -520000000000.0) {
		tmp = Math.sin((eps + x)) - Math.sin(x);
	} else if (eps <= 6e-6) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -520000000000.0:
		tmp = math.sin((eps + x)) - math.sin(x)
	elif eps <= 6e-6:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -520000000000.0)
		tmp = Float64(sin(Float64(eps + x)) - sin(x));
	elseif (eps <= 6e-6)
		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 <= -520000000000.0)
		tmp = sin((eps + x)) - sin(x);
	elseif (eps <= 6e-6)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -520000000000.0], N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6e-6], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.2e11

   1. Initial program 53.4%

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

   if -5.2e11 < eps < 6.0000000000000002e-6

   1. Initial program 25.6%

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

   2. Taylor expanded in eps around 0 98.1%

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

   if 6.0000000000000002e-6 < eps

   1. Initial program 56.6%

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

   2. Taylor expanded in x around 0 57.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -520000000000:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -520000000000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -520000000000.0) (not (<= eps 1.55e-6)))
   (sin eps)
   (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -520000000000.0) || !(eps <= 1.55e-6)) {
		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 <= (-520000000000.0d0)) .or. (.not. (eps <= 1.55d-6))) 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 <= -520000000000.0) || !(eps <= 1.55e-6)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -520000000000.0) or not (eps <= 1.55e-6):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -520000000000.0) || !(eps <= 1.55e-6))
		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 <= -520000000000.0) || ~((eps <= 1.55e-6)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -520000000000.0], N[Not[LessEqual[eps, 1.55e-6]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.2e11 or 1.55e-6 < eps

   1. Initial program 54.9%

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

   2. Taylor expanded in x around 0 55.3%

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

   if -5.2e11 < eps < 1.55e-6

   1. Initial program 25.6%

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

   2. Taylor expanded in eps around 0 98.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -520000000000 \lor \neg \left(\varepsilon \leq 1.55 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 8: 55.0% 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 41.8%

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

2. Taylor expanded in x around 0 53.8%

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

3. Final simplification 53.8%

   \[\leadsto \sin \varepsilon \]

Alternative 9: 29.4% 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 41.8%

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

2. Taylor expanded in eps around 0 45.6%

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

3. Taylor expanded in x around 0 24.9%

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

4. Final simplification 24.9%

   \[\leadsto \varepsilon \]

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023310 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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