Maksimov and Kolovsky, Equation (4)

Average Accuracy: 37.8% → 52.6%

Time: 8.8s

Precision: binary64

Cost: 7104

Math

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

↓

\[\left(J + J\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \ell\right) + U \]

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 (+ (* (+ J J) (* (cos (* K 0.5)) l)) U))

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 ((J + J) * (cos((K * 0.5)) * l)) + U;
}

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 = ((j + j) * (cos((k * 0.5d0)) * l)) + u
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 ((J + J) * (Math.cos((K * 0.5)) * l)) + U;
}

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 ((J + J) * (math.cos((K * 0.5)) * l)) + U

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(Float64(Float64(J + J) * Float64(cos(Float64(K * 0.5)) * l)) + U)
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 = ((J + J) * (cos((K * 0.5)) * l)) + U;
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[(N[(N[(J + J), $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 30.0%

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

2. Taylor expanded in l around 0 45.2%

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

3. Simplified 45.2%

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

4. Applied egg-rr 29.3%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\ell \cdot \left(\left(J + J\right) \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)} + U \]

5. Applied egg-rr 45.3%

   \[\leadsto \color{blue}{\left(J + J\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \ell\right)} + U \]

6. Final simplification 45.3%

   \[\leadsto \left(J + J\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \ell\right) + U \]

Alternatives

Alternative 1
Accuracy	52.6%
Cost	7104

\[U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right) \]

Alternative 2
Accuracy	45.7%
Cost	448

\[U + \left(J + J\right) \cdot \ell \]

Alternative 3
Accuracy	38.0%
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))