2sin (example 3.3)

Average Error: 37.2 → 0.2

Time: 1.5min

Precision: binary64

Cost: 32836

Math

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

↓

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

FPCore

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

↓

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

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(eps) * cos(x)) - (sin(x) * tmp)
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(eps) * Math.cos(x)) - (Math.sin(x) * tmp);
}

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(eps) * math.cos(x)) - (math.sin(x) * tmp)

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(sin(eps) * cos(x)) - Float64(sin(x) * tmp))
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(eps) * cos(x)) - (sin(x) * tmp);
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[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - 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]), $MachinePrecision]

Target

Original	37.2
Target	15.4
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 37.2

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

2. Applied egg-rr 0.4

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

3. Applied egg-rr 0.4

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

4. Simplified 0.4

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

5. Taylor expanded in eps around 0 0.4

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

6. Applied egg-rr 0.2

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

Alternatives

Alternative 1
Error	0.4
Cost	32448

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

Alternative 2
Error	0.4
Cost	26176

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

Alternative 3
Error	14.9
Cost	13640

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.026:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00034:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + -0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	14.7
Cost	13640

\[\begin{array}{l} t_0 := \left(\left(\sin \left(x + \varepsilon\right) + x\right) - x\right) - \sin x\\ \mathbf{if}\;\varepsilon \leq -0.0007:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.00055:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + -0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.5
Cost	13632

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

Alternative 6
Error	15.1
Cost	13256

\[\begin{array}{l} t_0 := \sin \varepsilon - \sin x\\ \mathbf{if}\;\varepsilon \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.000112:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.5
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000105:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8
Error	28.9
Cost	6464

\[\sin \varepsilon \]

Alternative 9
Error	45.4
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023033 
(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)))