2sin (example 3.3)

Average Accuracy: 42.4% → 99.6%

Time: 17.0s

Precision: binary64

Cost: 32832

Math

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

↓

\[\sin \varepsilon \cdot \cos x + \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin x \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (+
  (* (sin eps) (cos x))
  (* (* (/ (sin eps) -1.0) (tan (/ eps 2.0))) (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)) + (((sin(eps) / -1.0) * tan((eps / 2.0))) * 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)) + (((sin(eps) / (-1.0d0)) * tan((eps / 2.0d0))) * 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.sin(eps) / -1.0) * Math.tan((eps / 2.0))) * 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.sin(eps) / -1.0) * math.tan((eps / 2.0))) * 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(Float64(Float64(sin(eps) / -1.0) * tan(Float64(eps / 2.0))) * 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)) + (((sin(eps) / -1.0) * tan((eps / 2.0))) * 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[(N[(N[Sin[eps], $MachinePrecision] / -1.0), $MachinePrecision] * N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	42.4%
Target	75.8%
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 39.5%

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

2. Applied egg-rr 62.5%

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

   [Start] 39.5	\[ \sin \left(x + \varepsilon\right) - \sin x \]
   sin-sum [=>] 62.4	\[ \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x \]
   associate--l+ [=>] 62.5	\[ \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \]

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] 62.5	\[ \sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right) \]
   +-commutative [=>] 62.5	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon} \]
   sub-neg [=>] 62.5	\[ \color{blue}{\left(\cos x \cdot \sin \varepsilon + \left(-\sin x\right)\right)} + \sin x \cdot \cos \varepsilon \]
   associate-+l+ [=>] 99.3	\[ \color{blue}{\cos x \cdot \sin \varepsilon + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right)} \]
   *-commutative [=>] 99.3	\[ \color{blue}{\sin \varepsilon \cdot \cos x} + \left(\left(-\sin x\right) + \sin x \cdot \cos \varepsilon\right) \]
   neg-mul-1 [=>] 99.3	\[ \sin \varepsilon \cdot \cos x + \left(\color{blue}{-1 \cdot \sin x} + \sin x \cdot \cos \varepsilon\right) \]
   *-commutative [=>] 99.3	\[ \sin \varepsilon \cdot \cos x + \left(-1 \cdot \sin x + \color{blue}{\cos \varepsilon \cdot \sin x}\right) \]
   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 99.4%

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

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

5. Simplified 99.6%

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

   [Start] 99.4	\[ \sin \varepsilon \cdot \cos x + \frac{\sin x \cdot \left(-{\sin \varepsilon}^{2}\right)}{\cos \varepsilon + 1} \]
   *-commutative [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{\left(-{\sin \varepsilon}^{2}\right) \cdot \sin x}}{\cos \varepsilon + 1} \]
   associate-/l* [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\frac{-{\sin \varepsilon}^{2}}{\frac{\cos \varepsilon + 1}{\sin x}}} \]
   associate-/r/ [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1} \cdot \sin x} \]
   *-lft-identity [<=] 99.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(1 \cdot \frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}\right)} \cdot \sin x \]
   metadata-eval [<=] 99.4	\[ \sin \varepsilon \cdot \cos x + \left(\color{blue}{\frac{-1}{-1}} \cdot \frac{-{\sin \varepsilon}^{2}}{\cos \varepsilon + 1}\right) \cdot \sin x \]
   times-frac [<=] 99.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\frac{-1 \cdot \left(-{\sin \varepsilon}^{2}\right)}{-1 \cdot \left(\cos \varepsilon + 1\right)}} \cdot \sin x \]
   neg-mul-1 [<=] 99.4	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{-\left(-{\sin \varepsilon}^{2}\right)}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \cdot \sin x \]
   remove-double-neg [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{{\sin \varepsilon}^{2}}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \cdot \sin x \]
   unpow2 [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{-1 \cdot \left(\cos \varepsilon + 1\right)} \cdot \sin x \]
   times-frac [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \color{blue}{\left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\cos \varepsilon + 1}\right)} \cdot \sin x \]
   +-commutative [=>] 99.4	\[ \sin \varepsilon \cdot \cos x + \left(\frac{\sin \varepsilon}{-1} \cdot \frac{\sin \varepsilon}{\color{blue}{1 + \cos \varepsilon}}\right) \cdot \sin x \]
   hang-0p-tan [=>] 99.6	\[ \sin \varepsilon \cdot \cos x + \left(\frac{\sin \varepsilon}{-1} \cdot \color{blue}{\tan \left(\frac{\varepsilon}{2}\right)}\right) \cdot \sin x \]

6. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	32512

\[\sin \varepsilon \cdot \mathsf{fma}\left(\tan \left(\varepsilon \cdot 0.5\right), -\sin x, \cos x\right) \]

Alternative 2
Accuracy	99.7%
Cost	26176

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

Alternative 3
Accuracy	77.0%
Cost	12992

\[\sin \varepsilon \cdot \cos x \]

Alternative 4
Accuracy	75.7%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 5
Accuracy	54.8%
Cost	6464

\[\sin \varepsilon \]

Alternative 6
Accuracy	28.6%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023157 
(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)))