2sin (example 3.3)

Average Error: 37.4 → 0.2

Time: 19.9s

Precision: binary64

Cost: 32900

Math

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

↓

\[\left(-\sin x \cdot \begin{array}{l} \mathbf{if}\;2 \ne 0:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;1 - \cos \varepsilon\\ \end{array}\right) + \cos x \cdot \sin \varepsilon \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (+
  (-
   (*
    (sin x)
    (if (!= 2.0 0.0) (* (tan (/ eps 2.0)) (sin eps)) (- 1.0 (cos eps)))))
  (* (cos x) (sin eps))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (2.0 != 0.0) {
		tmp = tan((eps / 2.0)) * sin(eps);
	} else {
		tmp = 1.0 - cos(eps);
	}
	return -(sin(x) * tmp) + (cos(x) * 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
    real(8) :: tmp
    if (2.0d0 /= 0.0d0) then
        tmp = tan((eps / 2.0d0)) * sin(eps)
    else
        tmp = 1.0d0 - cos(eps)
    end if
    code = -(sin(x) * tmp) + (cos(x) * 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) {
	double tmp;
	if (2.0 != 0.0) {
		tmp = Math.tan((eps / 2.0)) * Math.sin(eps);
	} else {
		tmp = 1.0 - Math.cos(eps);
	}
	return -(Math.sin(x) * tmp) + (Math.cos(x) * Math.sin(eps));
}

Python

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

↓

def code(x, eps):
	tmp = 0
	if 2.0 != 0.0:
		tmp = math.tan((eps / 2.0)) * math.sin(eps)
	else:
		tmp = 1.0 - math.cos(eps)
	return -(math.sin(x) * tmp) + (math.cos(x) * math.sin(eps))

Julia

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

↓

function code(x, eps)
	tmp = 0.0
	if (2.0 != 0.0)
		tmp = Float64(tan(Float64(eps / 2.0)) * sin(eps));
	else
		tmp = Float64(1.0 - cos(eps));
	end
	return Float64(Float64(-Float64(sin(x) * tmp)) + Float64(cos(x) * sin(eps)))
end

MATLAB

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

↓

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (2.0 ~= 0.0)
		tmp = tan((eps / 2.0)) * sin(eps);
	else
		tmp = 1.0 - cos(eps);
	end
	tmp_2 = -(sin(x) * tmp) + (cos(x) * 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[(N[Sin[x], $MachinePrecision] * If[Unequal[2.0, 0.0], N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	37.4
Target	15.2
Herbie	0.2

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

Derivation

1. Initial program 37.4

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

2. Applied egg-rr 0.4

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

3. Simplified 0.4

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

4. Applied egg-rr 0.3

   \[\leadsto \left(-\sin x \cdot \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;1 + \cos \varepsilon \ne 0:\\ \;\;\;\;\frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;1 - \cos \varepsilon\\ } \end{array}}\right) + \cos x \cdot \sin \varepsilon \]

5. Taylor expanded in eps around 0 0.3

   \[\leadsto \left(-\sin x \cdot \begin{array}{l} \mathbf{if}\;\color{blue}{2} \ne 0:\\ \;\;\;\;\frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;1 - \cos \varepsilon\\ \end{array}\right) + \cos x \cdot \sin \varepsilon \]

6. Applied egg-rr 0.2

   \[\leadsto \left(-\sin x \cdot \begin{array}{l} \mathbf{if}\;2 \ne 0:\\ \;\;\;\;\color{blue}{\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;1 - \cos \varepsilon\\ \end{array}\right) + \cos x \cdot \sin \varepsilon \]

Alternatives

Alternative 1
Error	0.4
Cost	32448

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

Alternative 2
Error	0.4
Cost	26176

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

Alternative 3
Error	15.2
Cost	13632

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

Alternative 4
Error	15.2
Cost	13632

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

Alternative 5
Error	15.1
Cost	13256

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon - \sin x\\ \end{array} \]

Alternative 6
Error	15.3
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7
Error	29.1
Cost	6464

\[\sin \varepsilon \]

Alternative 8
Error	45.4
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023010 
(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)))