Maksimov and Kolovsky, Equation (4)

Average Error: 18.0 → 0.1

Time: 9.0s

Precision: binary64

Math

\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

↓

\[U + \left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

↓

(FPCore (J l K U)
 :precision binary64
 (+ U (* (* J (* 2.0 (sinh l))) (cos (/ K 2.0)))))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

↓

double code(double J, double l, double K, double U) {
	return U + ((J * (2.0 * sinh(l))) * cos((K / 2.0)));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

↓

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + ((j * (2.0d0 * sinh(l))) * cos((k / 2.0d0)))
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

↓

public static double code(double J, double l, double K, double U) {
	return U + ((J * (2.0 * Math.sinh(l))) * Math.cos((K / 2.0)));
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

↓

def code(J, l, K, U):
	return U + ((J * (2.0 * math.sinh(l))) * math.cos((K / 2.0)))

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

↓

function code(J, l, K, U)
	return Float64(U + Float64(Float64(J * Float64(2.0 * sinh(l))) * cos(Float64(K / 2.0))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

↓

function tmp = code(J, l, K, U)
	tmp = U + ((J * (2.0 * sinh(l))) * cos((K / 2.0)));
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

↓

code[J_, l_, K_, U_] := N[(U + N[(N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

1. Initial program 18.0

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

2. Applied sinh-undef_binary64 0.1

   \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \sinh \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

3. Applied +-commutative_binary64 0.1

   \[\leadsto \color{blue}{U + \left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)} \]

4. Applied pow1_binary64 0.1

   \[\leadsto U + \left(J \cdot \left(2 \cdot \color{blue}{{\sinh \ell}^{1}}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \]

5. Applied pow1_binary64 0.1

   \[\leadsto U + \left(J \cdot \left(\color{blue}{{2}^{1}} \cdot {\sinh \ell}^{1}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \]

6. Applied pow-prod-down_binary64 0.1

   \[\leadsto U + \left(J \cdot \color{blue}{{\left(2 \cdot \sinh \ell\right)}^{1}}\right) \cdot \cos \left(\frac{K}{2}\right) \]

7. Applied pow1_binary64 0.1

   \[\leadsto U + \left(\color{blue}{{J}^{1}} \cdot {\left(2 \cdot \sinh \ell\right)}^{1}\right) \cdot \cos \left(\frac{K}{2}\right) \]

8. Applied pow-prod-down_binary64 0.1

   \[\leadsto U + \color{blue}{{\left(J \cdot \left(2 \cdot \sinh \ell\right)\right)}^{1}} \cdot \cos \left(\frac{K}{2}\right) \]

9. Final simplification 0.1

   \[\leadsto U + \left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) \]

Reproduce

herbie shell --seed 2022131 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))