Average Error: 39.9 → 0.3
Time: 12.8s
Precision: binary64
Cost: 26240
\[\cos \left(x + \varepsilon\right) - \cos x \]
↓
\[\sin \varepsilon \cdot \left(\left(-\sin x\right) - \cos x \cdot \tan \left(\varepsilon \cdot 0.5\right)\right) \]
(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))
↓
(FPCore (x eps)
 :precision binary64
 (* (sin eps) (- (- (sin x)) (* (cos x) (tan (* eps 0.5))))))
double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}
↓
double code(double x, double eps) {
	return sin(eps) * (-sin(x) - (cos(x) * tan((eps * 0.5))));
}

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 = sin(eps) * (-sin(x) - (cos(x) * tan((eps * 0.5d0))))
end function

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.sin(eps) * (-Math.sin(x) - (Math.cos(x) * Math.tan((eps * 0.5))));
}

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)
↓
def code(x, eps):
	return math.sin(eps) * (-math.sin(x) - (math.cos(x) * math.tan((eps * 0.5))))

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end
↓
function code(x, eps)
	return Float64(sin(eps) * Float64(Float64(-sin(x)) - Float64(cos(x) * tan(Float64(eps * 0.5)))))
end

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end
↓
function tmp = code(x, eps)
	tmp = sin(eps) * (-sin(x) - (cos(x) * tan((eps * 0.5))));
end

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[((-N[Sin[x], $MachinePrecision]) - N[(N[Cos[x], $MachinePrecision] * N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \varepsilon \cdot \left(\left(-\sin x\right) - \cos x \cdot \tan \left(\varepsilon \cdot 0.5\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.9

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

  2. Applied egg-rr24.2

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

  3. Taylor expanded in x around inf 24.2

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

  4. Simplified6.6

    \[\leadsto \color{blue}{\cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x} \]
    Proof
    (-.f64 (*.f64 (cos.f64 x) (+.f64 (cos.f64 eps) -1)) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (cos.f64 x) (+.f64 (cos.f64 eps) (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (*.f64 (cos.f64 x) (Rewrite<= sub-neg_binary64 (-.f64 (cos.f64 eps) 1))) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (*.f64 (cos.f64 x) 1))) (*.f64 (sin.f64 eps) (sin.f64 x))): 13 points increase in error, 5 points decrease in error
    (-.f64 (-.f64 (*.f64 (cos.f64 x) (cos.f64 eps)) (Rewrite=> *-rgt-identity_binary64 (cos.f64 x))) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 eps) (cos.f64 x))) (cos.f64 x)) (*.f64 (sin.f64 eps) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (-.f64 (-.f64 (*.f64 (cos.f64 eps) (cos.f64 x)) (cos.f64 x)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 x) (sin.f64 eps)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 (cos.f64 eps) (cos.f64 x)) (+.f64 (cos.f64 x) (*.f64 (sin.f64 x) (sin.f64 eps))))): 102 points increase in error, 20 points decrease in error

  5. Applied egg-rr0.7

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

  6. Simplified0.3

    \[\leadsto \cos x \cdot \color{blue}{\left(-\tan \left(\frac{\varepsilon}{2}\right) \cdot \sin \varepsilon\right)} - \sin \varepsilon \cdot \sin x \]
    Proof
    (neg.f64 (*.f64 (tan.f64 (/.f64 eps 2)) (sin.f64 eps))): 0 points increase in error, 0 points decrease in error
    (neg.f64 (*.f64 (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 eps) (+.f64 1 (cos.f64 eps)))) (sin.f64 eps))): 38 points increase in error, 21 points decrease in error
    (neg.f64 (*.f64 (/.f64 (sin.f64 eps) (Rewrite<= +-commutative_binary64 (+.f64 (cos.f64 eps) 1))) (sin.f64 eps))): 0 points increase in error, 0 points decrease in error
    (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (sin.f64 eps) (/.f64 (+.f64 (cos.f64 eps) 1) (sin.f64 eps))))): 43 points increase in error, 36 points decrease in error
    (neg.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sin.f64 eps) (sin.f64 eps)) (+.f64 (cos.f64 eps) 1)))): 31 points increase in error, 42 points decrease in error
    (neg.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (+.f64 (cos.f64 eps) 1))): 0 points increase in error, 0 points decrease in error
    (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) 1)) (+.f64 (cos.f64 eps) 1)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 (pow.f64 (sin.f64 eps) 2)) (/.f64 1 (+.f64 (cos.f64 eps) 1)))): 19 points increase in error, 23 points decrease in error

  7. Applied egg-rr0.3

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

  8. Simplified0.3

    \[\leadsto \color{blue}{\left(-\sin \varepsilon\right) \cdot \left(\cos x \cdot \tan \left(0.5 \cdot \varepsilon\right) + \sin x\right)} \]
    Proof
    (*.f64 (neg.f64 (sin.f64 eps)) (+.f64 (*.f64 (cos.f64 x) (tan.f64 (*.f64 1/2 eps))) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (*.f64 (neg.f64 (sin.f64 eps)) (+.f64 (*.f64 (cos.f64 x) (tan.f64 (Rewrite=> *-commutative_binary64 (*.f64 eps 1/2)))) (sin.f64 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (cos.f64 x) (tan.f64 (*.f64 eps 1/2))) (neg.f64 (sin.f64 eps))) (*.f64 (sin.f64 x) (neg.f64 (sin.f64 eps))))): 17 points increase in error, 15 points decrease in error
    (+.f64 (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (*.f64 (cos.f64 x) (tan.f64 (*.f64 eps 1/2))) (sin.f64 eps)))) (*.f64 (sin.f64 x) (neg.f64 (sin.f64 eps)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps))))) (*.f64 (sin.f64 x) (neg.f64 (sin.f64 eps)))): 6 points increase in error, 6 points decrease in error
    (+.f64 (neg.f64 (*.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)))) (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (sin.f64 x) (sin.f64 eps))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (*.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)))) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 eps) (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> distribute-neg-out_binary64 (neg.f64 (+.f64 (*.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps))) (*.f64 (sin.f64 eps) (sin.f64 x))))): 0 points increase in error, 0 points decrease in error
    (neg.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x))))): 7 points increase in error, 8 points decrease in error
    (Rewrite<= +-rgt-identity_binary64 (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) 0)): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (Rewrite<= metadata-eval (log.f64 1))): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (Rewrite<= lft-mult-inverse_binary64 (*.f64 (/.f64 1 (pow.f64 (exp.f64 (sin.f64 x)) (sin.f64 eps))) (pow.f64 (exp.f64 (sin.f64 x)) (sin.f64 eps)))))): 12 points increase in error, 2 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (*.f64 (/.f64 1 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (sin.f64 x) (sin.f64 eps))))) (pow.f64 (exp.f64 (sin.f64 x)) (sin.f64 eps))))): 13 points increase in error, 1 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (*.f64 (/.f64 1 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 eps) (sin.f64 x))))) (pow.f64 (exp.f64 (sin.f64 x)) (sin.f64 eps))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (*.f64 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 (*.f64 (sin.f64 eps) (sin.f64 x))))) (pow.f64 (exp.f64 (sin.f64 x)) (sin.f64 eps))))): 10 points increase in error, 9 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (*.f64 (exp.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 (sin.f64 eps) (neg.f64 (sin.f64 x))))) (pow.f64 (exp.f64 (sin.f64 x)) (sin.f64 eps))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (*.f64 (exp.f64 (*.f64 (sin.f64 eps) (neg.f64 (sin.f64 x)))) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (sin.f64 x) (sin.f64 eps))))))): 4 points increase in error, 10 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (*.f64 (exp.f64 (*.f64 (sin.f64 eps) (neg.f64 (sin.f64 x)))) (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 eps) (sin.f64 x))))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 (sin.f64 eps) (neg.f64 (sin.f64 x))) (*.f64 (sin.f64 eps) (sin.f64 x))))))): 1 points increase in error, 18 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (log.f64 (exp.f64 (Rewrite<= fma-udef_binary64 (fma.f64 (sin.f64 eps) (neg.f64 (sin.f64 x)) (*.f64 (sin.f64 eps) (sin.f64 x))))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (neg.f64 (fma.f64 (cos.f64 x) (*.f64 (tan.f64 (*.f64 eps 1/2)) (sin.f64 eps)) (*.f64 (sin.f64 eps) (sin.f64 x)))) (Rewrite=> rem-log-exp_binary64 (fma.f64 (sin.f64 eps) (neg.f64 (sin.f64 x)) (*.f64 (sin.f64 eps) (sin.f64 x))))): 8 points increase in error, 4 points decrease in error

  9. Final simplification0.3

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

Alternatives

Alternative 1
Error0.8
Cost26440
\[\begin{array}{l} t_0 := \cos x \cdot \left(\cos \varepsilon + -1\right) - \sin \varepsilon \cdot \sin x\\ t_1 := \sin \left(\varepsilon \cdot 0.5\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-93}:\\ \;\;\;\;\left(x \cdot \cos \left(\varepsilon \cdot 0.5\right) + t_1\right) \cdot \left(t_1 \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error15.1
Cost13640
\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0155:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.011:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\cos x \cdot -0.5\right) - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error15.5
Cost13632
\[\left(\sin \left(\varepsilon \cdot 0.5\right) \cdot -2\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right) \]
Alternative 4
Error15.3
Cost13256
\[\begin{array}{l} t_0 := \cos \varepsilon - \cos x\\ \mathbf{if}\;\varepsilon \leq -0.0034:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0023:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error15.7
Cost7240
\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.0044:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 0.0052:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error22.1
Cost7184
\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \sin x \cdot \left(-\varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 8.8 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 1.06 \cdot 10^{-79}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \varepsilon \cdot x\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error30.6
Cost6856
\[\begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -0.000162:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 9.8 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error46.8
Cost576
\[\varepsilon \cdot \left(\varepsilon \cdot -0.5\right) - \varepsilon \cdot x \]
Alternative 9
Error50.0
Cost320
\[-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) \]
Alternative 10
Error52.7
Cost256
\[\varepsilon \cdot \left(-x\right) \]
Alternative 11
Error55.6
Cost64
\[0 \]

Error

Reproduce

herbie shell --seed 2022330 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))