2sin (example 3.3)

Average Error: 36.6 → 0.4

Time: 8.2s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

↓

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)) / sin(eps))))
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	15.0
Herbie	0.4

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

Derivation

1. Initial program 36.6

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

2. Applied egg-rr 21.5

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

3. Taylor expanded in x around inf 21.4

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

4. Simplified 0.4

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

5. Applied egg-rr 0.4

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

6. Taylor expanded in eps around inf 0.4

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

7. Simplified 0.4

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

8. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022133 
(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)))