(FPCore (alpha beta i)
:precision binary64
(/
(+
(/
(/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
(+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
1.0)
2.0))(FPCore (alpha beta i)
:precision binary64
(let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
(if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.999998)
(* 0.5 (/ (+ (+ 2.0 (* beta 2.0)) (* i 4.0)) alpha))
(/
(fma
(/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))
(/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
1.0)
2.0))))double code(double alpha, double beta, double i) {
return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
}
double code(double alpha, double beta, double i) {
double t_0 = (alpha + beta) + (2.0 * i);
double tmp;
if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999998) {
tmp = 0.5 * (((2.0 + (beta * 2.0)) + (i * 4.0)) / alpha);
} else {
tmp = fma(((alpha + beta) / (beta + fma(2.0, i, alpha))), ((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))), 1.0) / 2.0;
}
return tmp;
}
function code(alpha, beta, i) return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / Float64(Float64(alpha + beta) + Float64(2.0 * i))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) + 2.0)) + 1.0) / 2.0) end
function code(alpha, beta, i) t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i)) tmp = 0.0 if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.999998) tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(i * 4.0)) / alpha)); else tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha))), Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 1.0) / 2.0); end return tmp end
code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(0.5 * N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
\begin{array}{l}
t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\
\mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999998:\\
\;\;\;\;0.5 \cdot \frac{\left(2 + \beta \cdot 2\right) + i \cdot 4}{\alpha}\\
\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\
\end{array}
if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999998000000000054Initial program 62.3
Taylor expanded in i around 0 61.8
Simplified59.5
Taylor expanded in alpha around inf 57.2
Simplified57.2
Taylor expanded in alpha around inf 5.9
if -0.999998000000000054 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) Initial program 12.7
Simplified0.0
Final simplification1.4
| Alternative 1 | |
|---|---|
| Error | 1.4 |
| Cost | 9796 |
| Alternative 2 | |
|---|---|
| Error | 2.3 |
| Cost | 2500 |
| Alternative 3 | |
|---|---|
| Error | 12.1 |
| Cost | 1228 |
| Alternative 4 | |
|---|---|
| Error | 13.8 |
| Cost | 1101 |
| Alternative 5 | |
|---|---|
| Error | 13.8 |
| Cost | 973 |
| Alternative 6 | |
|---|---|
| Error | 15.0 |
| Cost | 972 |
| Alternative 7 | |
|---|---|
| Error | 15.9 |
| Cost | 708 |
| Alternative 8 | |
|---|---|
| Error | 18.2 |
| Cost | 196 |
| Alternative 9 | |
|---|---|
| Error | 43.2 |
| Cost | 64 |
herbie shell --seed 2022343
(FPCore (alpha beta i)
:name "Octave 3.8, jcobi/2"
:precision binary64
:pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
(/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))