2sin (example 3.3)

Percentage Accurate: 41.4% → 99.6%

Time: 17.7s

Precision: binary64

Cost: 32704

Math

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

↓

\[\sin \varepsilon \cdot \cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \varepsilon \cdot \sin x\right) \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (- (* (sin eps) (cos x)) (* (tan (* eps 0.5)) (* (sin eps) (sin x)))))

C

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

↓

double code(double x, double eps) {
	return (sin(eps) * cos(x)) - (tan((eps * 0.5)) * (sin(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

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) - (tan((eps * 0.5d0)) * (sin(eps) * sin(x)))
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.tan((eps * 0.5)) * (Math.sin(eps) * Math.sin(x)));
}

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.tan((eps * 0.5)) * (math.sin(eps) * math.sin(x)))

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(tan(Float64(eps * 0.5)) * Float64(sin(eps) * sin(x))))
end

MATLAB

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

↓

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) - (tan((eps * 0.5)) * (sin(eps) * sin(x)));
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[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	41.4%
Target	75.9%
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 41.6%

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

2. Applied egg-rr 68.9%

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

   [Start] 41.6%	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sin-sum [=>] 69.0%	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]
   associate--l+ [=>] 68.9%	\[ \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

3. Simplified 99.4%

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

   [Start] 68.9%	\[ \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \]
   +-commutative [=>] 68.9%	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon} \]
   sub-neg [=>] 68.9%	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} + \sin x \cdot \cos \varepsilon \]
   associate-+l+ [=>] 99.4%	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} \]
   *-commutative [=>] 99.4%	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right) \]
   neg-mul-1 [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \left(\color{blue}{-1 \cdot \sin x} + \sin x \cdot \cos \varepsilon\right) \]
   *-commutative [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \left(-1 \cdot \sin x + \color{blue}{\cos \varepsilon \cdot \sin x}\right) \]
   distribute-rgt-out [=>] 99.4%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\sin x \cdot \left(-1 + \cos \varepsilon\right)} \]
   +-commutative [<=] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\cos \varepsilon + -1\right)} \]

4. Applied egg-rr 99.2%

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

   [Start] 99.4%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
   flip-+ [=>] 99.1%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1}{\cos \varepsilon - -1}} \]
   frac-2neg [=>] 99.1%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\frac{-\left(\cos \varepsilon \cdot \cos \varepsilon - -1 \cdot -1\right)}{-\left(\cos \varepsilon - -1\right)}} \]
   metadata-eval [=>] 99.1%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(\cos \varepsilon \cdot \cos \varepsilon - \color{blue}{1}\right)}{-\left(\cos \varepsilon - -1\right)} \]
   sub-1-cos [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\color{blue}{\left(-\sin \varepsilon \cdot \sin \varepsilon\right)}}{-\left(\cos \varepsilon - -1\right)} \]
   pow2 [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-\color{blue}{{\sin \varepsilon}^{2}}\right)}{-\left(\cos \varepsilon - -1\right)} \]
   sub-neg [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\color{blue}{\left(\cos \varepsilon + \left(--1\right)\right)}} \]
   metadata-eval [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\cos \varepsilon + \color{blue}{1}\right)} \]

5. Simplified 99.6%

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

   [Start] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\cos \varepsilon + 1\right)} \]
   remove-double-neg [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{-\left(\cos \varepsilon + 1\right)} \]
   unpow2 [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-\left(\cos \varepsilon + 1\right)} \]
   neg-mul-1 [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\color{blue}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \]
   times-frac [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \]
   +-commutative [=>] 99.2%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \]
   hang-0p-tan [=>] 99.6%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \]

6. Applied egg-rr 95.0%

   \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(-\sin \varepsilon\right)\right) \cdot \sin x\right)} - 1\right)} \]
   Step-by-step derivation

   [Start] 99.6%	\[ \sin \varepsilon \cdot \cos x + \sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \]
   expm1-log1p-u [=>] 95.4%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)\right)} \]
   expm1-udef [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)} - 1\right)} \]
   *-commutative [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x}\right)} - 1\right) \]
   *-commutative [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\tan \left(\frac{\varepsilon}{2}\right) \cdot \frac{\sin \varepsilon}{-1}\right)} \cdot \sin x\right)} - 1\right) \]
   div-inv [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\left(\tan \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \frac{\sin \varepsilon}{-1}\right) \cdot \sin x\right)} - 1\right) \]
   metadata-eval [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\left(\tan \left(\varepsilon \cdot \color{blue}{0.5}\right) \cdot \frac{\sin \varepsilon}{-1}\right) \cdot \sin x\right)} - 1\right) \]
   frac-2neg [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \color{blue}{\frac{-\sin \varepsilon}{--1}}\right) \cdot \sin x\right)} - 1\right) \]
   metadata-eval [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \frac{-\sin \varepsilon}{\color{blue}{1}}\right) \cdot \sin x\right)} - 1\right) \]
   /-rgt-identity [=>] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \color{blue}{\left(-\sin \varepsilon\right)}\right) \cdot \sin x\right)} - 1\right) \]

7. Simplified 99.7%

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

   [Start] 95.0%	\[ \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(-\sin \varepsilon\right)\right) \cdot \sin x\right)} - 1\right) \]
   expm1-def [=>] 95.4%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(-\sin \varepsilon\right)\right) \cdot \sin x\right)\right)} \]
   expm1-log1p [=>] 99.6%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(-\sin \varepsilon\right)\right) \cdot \sin x} \]
   associate-*l* [=>] 99.7%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\left(-\sin \varepsilon\right) \cdot \sin x\right)} \]
   distribute-lft-neg-in [<=] 99.7%	\[ \sin \varepsilon \cdot \cos x + \tan \left(\varepsilon \cdot 0.5\right) \cdot \color{blue}{\left(-\sin \varepsilon \cdot \sin x\right)} \]

8. Applied egg-rr 99.7%

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

   [Start] 99.7%	\[ \sin \varepsilon \cdot \cos x + \tan \left(\varepsilon \cdot 0.5\right) \cdot \left(-\sin \varepsilon \cdot \sin x\right) \]
   distribute-rgt-neg-out [=>] 99.7%	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(-\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right)} \]
   unsub-neg [=>] 99.7%	\[ \color{blue}{\sin \varepsilon \cdot \cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)} \]

9. Simplified 99.7%

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

   [Start] 99.7%	\[ \sin \varepsilon \cdot \cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \varepsilon \cdot \sin x\right) \]
   *-commutative [=>] 99.7%	\[ \sin \varepsilon \cdot \cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \color{blue}{\left(\sin x \cdot \sin \varepsilon\right)} \]

10. Final simplification 99.7%

    \[\leadsto \sin \varepsilon \cdot \cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \varepsilon \cdot \sin x\right) \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	32704

\[\sin \varepsilon \cdot \cos x - \tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin \varepsilon \cdot \sin x\right) \]

Alternative 2
Accuracy	99.3%
Cost	32576

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

Alternative 3
Accuracy	99.3%
Cost	32448

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

Alternative 4
Accuracy	99.3%
Cost	26176

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

Alternative 5
Accuracy	76.7%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00032 \lor \neg \left(\varepsilon \leq 0.00175\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \end{array} \]

Alternative 6
Accuracy	75.9%
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	76.4%
Cost	13257

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00012 \lor \neg \left(\varepsilon \leq 0.00036\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 8
Accuracy	75.8%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9
Accuracy	54.7%
Cost	6464

\[\sin \varepsilon \]

Alternative 10
Accuracy	4.3%
Cost	64

\[0 \]

Alternative 11
Accuracy	29.3%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023263 
(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)))