2sin (example 3.3)

Percentage Accurate: 41.5% → 99.6%

Time: 19.4s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

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

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) * (sin(eps) * tan((eps / 2.0))));
}

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) * (sin(eps) * tan((eps / 2.0d0))))
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) * (Math.sin(eps) * Math.tan((eps / 2.0))));
}

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) * (math.sin(eps) * math.tan((eps / 2.0))))

Julia

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

↓

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) - Float64(sin(x) * Float64(sin(eps) * tan(Float64(eps / 2.0)))))
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) * (sin(eps) * tan((eps / 2.0))));
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] * N[(N[Sin[eps], $MachinePrecision] * N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	41.5%
Target	76.6%
Herbie	99.6%

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

Derivation

1. Initial program 36.3%

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

2. Applied egg-rr 63.4%

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

   [Start] 36.3%	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sin-sum [=>] 63.4%	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]
   associate--l+ [=>] 63.4%	\[ \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

3. Simplified 99.1%

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

   [Start] 63.4%	\[ \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \]
   +-commutative [=>] 63.4%	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon} \]
   sub-neg [=>] 63.4%	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} + \sin x \cdot \cos \varepsilon \]
   associate-+l+ [=>] 99.1%	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} \]
   *-commutative [=>] 99.1%	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right) \]
   neg-mul-1 [=>] 99.1%	\[ \sin \varepsilon \cdot \cos x + \left(\color{blue}{-1 \cdot \sin x} + \sin x \cdot \cos \varepsilon\right) \]
   *-commutative [=>] 99.1%	\[ \sin \varepsilon \cdot \cos x + \left(-1 \cdot \sin x + \color{blue}{\cos \varepsilon \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 99.1%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(-1 + \cos \varepsilon\right)} \]
   +-commutative [<=] 99.1%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} \]

4. Applied egg-rr 99.3%

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

   [Start] 99.1%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
   flip-+ [=>] 98.9%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1}{\cos \varepsilon - -1}} \]
   metadata-eval [=>] 98.9%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - \color{blue}{1}}{\cos \varepsilon - -1} \]
   sub-1-cos [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{-\sin \varepsilon \cdot \sin \varepsilon}}{\cos \varepsilon - -1} \]
   pow2 [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\color{blue}{{\sin \varepsilon}^{2}}}{\cos \varepsilon - -1} \]

5. Taylor expanded in eps around inf 99.3%

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

6. Simplified 99.6%

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

   [Start] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(-1 \cdot \frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right) \]
   mul-1-neg [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(-\frac{{\sin \varepsilon}^{2}}{1 + \cos \varepsilon}\right)} \]
   unpow2 [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(-\frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{1 + \cos \varepsilon}\right) \]
   +-commutative [<=] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(-\frac{\sin \varepsilon \cdot \sin \varepsilon}{\color{blue}{\cos \varepsilon + 1}}\right) \]
   associate-*r/ [<=] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(-\color{blue}{\sin \varepsilon \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}}\right) \]
   distribute-lft-neg-in [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\left(-\sin \varepsilon\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \]
   +-commutative [=>] 99.3%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\left(-\sin \varepsilon\right) \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \]
   hang-0p-tan [=>] 99.6%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\left(-\sin \varepsilon\right) \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \]

7. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	32704

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

Alternative 2
Accuracy	99.4%
Cost	32448

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

Alternative 3
Accuracy	99.4%
Cost	32448

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

Alternative 4
Accuracy	99.4%
Cost	26176

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

Alternative 5
Accuracy	76.2%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -190000000000:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.000245:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 6
Accuracy	76.6%
Cost	13632

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

Alternative 7
Accuracy	76.0%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -190000000000:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.00016:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8
Accuracy	76.6%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.009:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9
Accuracy	54.9%
Cost	6464

\[\sin \varepsilon \]

Alternative 10
Accuracy	4.3%
Cost	64

\[0 \]

Alternative 11
Accuracy	30.1%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023277 
(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)))