2cos (problem 3.3.5)

Percentage Accurate: 85.1% → 99.5%

Time: 1.5min

Precision: binary64

Cost: 39168

• could not determine a ground truth

  1. x = -1.1988207350358308e+257
  2. eps = -1.1554447212373164e-282

Math

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

↓

\[\cos x \cdot \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (-
  (* (cos x) (/ (pow (sin eps) 2.0) (- -1.0 (cos eps))))
  (* (sin eps) (sin x))))

C

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

↓

double code(double x, double eps) {
	return (cos(x) * (pow(sin(eps), 2.0) / (-1.0 - cos(eps)))) - (sin(eps) * sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos(x) * ((sin(eps) ** 2.0d0) / ((-1.0d0) - cos(eps)))) - (sin(eps) * sin(x))
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

↓

public static double code(double x, double eps) {
	return (Math.cos(x) * (Math.pow(Math.sin(eps), 2.0) / (-1.0 - Math.cos(eps)))) - (Math.sin(eps) * Math.sin(x));
}

Python

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

↓

def code(x, eps):
	return (math.cos(x) * (math.pow(math.sin(eps), 2.0) / (-1.0 - math.cos(eps)))) - (math.sin(eps) * math.sin(x))

Julia

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

↓

function code(x, eps)
	return Float64(Float64(cos(x) * Float64((sin(eps) ^ 2.0) / Float64(-1.0 - cos(eps)))) - Float64(sin(eps) * sin(x)))
end

MATLAB

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

↓

function tmp = code(x, eps)
	tmp = (cos(x) * ((sin(eps) ^ 2.0) / (-1.0 - cos(eps)))) - (sin(eps) * sin(x));
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.7%

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

2. Applied egg-rr 99.1%

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

   [Start] 83.7%	\[ \cos \left(x + \varepsilon\right) - \cos x \]
   sub-neg [=>] 83.7%	\[ \color{blue}{\cos \left(x + \varepsilon\right) + \left(-\cos x\right)} \]
   cos-sum [=>] 99.1%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} + \left(-\cos x\right) \]
   associate-+l- [=>] 99.1%	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)} \]
   fma-neg [=>] 99.1%	\[ \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right)} \]

3. Applied egg-rr 99.1%

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

   [Start] 99.1%	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon - \left(-\cos x\right)\right)\right) \]
   sub-neg [=>] 99.1%	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\left(\sin x \cdot \sin \varepsilon + \left(-\left(-\cos x\right)\right)\right)}\right) \]
   remove-double-neg [=>] 99.1%	\[ \mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon + \color{blue}{\cos x}\right)\right) \]
   fma-neg [<=] 99.1%	\[ \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
   fma-def [=>] 99.1%	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)} \]

4. Applied egg-rr 99.1%

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

   [Start] 99.1%	\[ \cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \]
   fma-udef [=>] 99.1%	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \]
   +-commutative [<=] 99.1%	\[ \cos x \cdot \cos \varepsilon - \color{blue}{\left(\cos x + \sin x \cdot \sin \varepsilon\right)} \]
   associate--r+ [=>] 99.1%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) - \sin x \cdot \sin \varepsilon} \]
   cancel-sign-sub-inv [=>] 99.1%	\[ \color{blue}{\left(\cos x \cdot \cos \varepsilon - \cos x\right) + \left(-\sin x\right) \cdot \sin \varepsilon} \]
   *-commutative [=>] 99.1%	\[ \left(\color{blue}{\cos \varepsilon \cdot \cos x} - \cos x\right) + \left(-\sin x\right) \cdot \sin \varepsilon \]
   *-un-lft-identity [=>] 99.1%	\[ \left(\cos \varepsilon \cdot \cos x - \color{blue}{1 \cdot \cos x}\right) + \left(-\sin x\right) \cdot \sin \varepsilon \]
   distribute-rgt-out-- [=>] 99.1%	\[ \color{blue}{\cos x \cdot \left(\cos \varepsilon - 1\right)} + \left(-\sin x\right) \cdot \sin \varepsilon \]

5. Applied egg-rr 99.5%

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

   [Start] 99.1%	\[ \cos x \cdot \left(\cos \varepsilon - 1\right) + \left(-\sin x\right) \cdot \sin \varepsilon \]
   flip-- [=>] 98.9%	\[ \cos x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   frac-2neg [=>] 98.9%	\[ \cos x \cdot \color{blue}{\frac{-\left(\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1\right)}{-\left(\cos \varepsilon + 1\right)}} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   metadata-eval [=>] 98.9%	\[ \cos x \cdot \frac{-\left(\cos \varepsilon \cdot \cos \varepsilon - \color{blue}{1}\right)}{-\left(\cos \varepsilon + 1\right)} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   sub-1-cos [=>] 99.5%	\[ \cos x \cdot \frac{-\color{blue}{\left(-\sin \varepsilon \cdot \sin \varepsilon\right)}}{-\left(\cos \varepsilon + 1\right)} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   pow2 [=>] 99.5%	\[ \cos x \cdot \frac{-\left(-\color{blue}{{\sin \varepsilon}^{2}}\right)}{-\left(\cos \varepsilon + 1\right)} + \left(-\sin x\right) \cdot \sin \varepsilon \]

6. Simplified 99.5%

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

   [Start] 99.5%	\[ \cos x \cdot \frac{-\left(-{\sin \varepsilon}^{2}\right)}{-\left(\cos \varepsilon + 1\right)} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   remove-double-neg [=>] 99.5%	\[ \cos x \cdot \frac{\color{blue}{{\sin \varepsilon}^{2}}}{-\left(\cos \varepsilon + 1\right)} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   neg-sub0 [=>] 99.5%	\[ \cos x \cdot \frac{{\sin \varepsilon}^{2}}{\color{blue}{0 - \left(\cos \varepsilon + 1\right)}} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   +-commutative [=>] 99.5%	\[ \cos x \cdot \frac{{\sin \varepsilon}^{2}}{0 - \color{blue}{\left(1 + \cos \varepsilon\right)}} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   associate--r+ [=>] 99.5%	\[ \cos x \cdot \frac{{\sin \varepsilon}^{2}}{\color{blue}{\left(0 - 1\right) - \cos \varepsilon}} + \left(-\sin x\right) \cdot \sin \varepsilon \]
   metadata-eval [=>] 99.5%	\[ \cos x \cdot \frac{{\sin \varepsilon}^{2}}{\color{blue}{-1} - \cos \varepsilon} + \left(-\sin x\right) \cdot \sin \varepsilon \]

7. Final simplification 99.5%

   \[\leadsto \cos x \cdot \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39168

\[\cos x \cdot \frac{{\sin \varepsilon}^{2}}{-1 - \cos \varepsilon} - \sin \varepsilon \cdot \sin x \]

Alternative 2
Accuracy	99.2%
Cost	32512

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

Alternative 3
Accuracy	99.2%
Cost	26176

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

Alternative 4
Accuracy	85.4%
Cost	13632

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

Alternative 5
Accuracy	85.1%
Cost	13120

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

Alternative 6
Accuracy	83.7%
Cost	12992

\[\cos \varepsilon - \cos x \]

Alternative 7
Accuracy	83.2%
Cost	6592

\[-1 + \cos \varepsilon \]

Alternative 8
Accuracy	74.1%
Cost	320

\[-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) \]

Alternative 9
Accuracy	72.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023243 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))