(FPCore (a b eps) :precision binary64 (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))
(FPCore (a b eps)
:precision binary64
(let* ((t_0 (/ (+ b a) a)))
(if (<= b -1.6e+17)
(* t_0 (/ eps (expm1 (* b eps))))
(if (<= b 3.3e-99)
(/ t_0 b)
(if (<= b 1.55e+227) (/ (/ (+ b a) b) a) (/ eps (expm1 (* a eps))))))))double code(double a, double b, double eps) {
return (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
}
double code(double a, double b, double eps) {
double t_0 = (b + a) / a;
double tmp;
if (b <= -1.6e+17) {
tmp = t_0 * (eps / expm1((b * eps)));
} else if (b <= 3.3e-99) {
tmp = t_0 / b;
} else if (b <= 1.55e+227) {
tmp = ((b + a) / b) / a;
} else {
tmp = eps / expm1((a * eps));
}
return tmp;
}
public static double code(double a, double b, double eps) {
return (eps * (Math.exp(((a + b) * eps)) - 1.0)) / ((Math.exp((a * eps)) - 1.0) * (Math.exp((b * eps)) - 1.0));
}
public static double code(double a, double b, double eps) {
double t_0 = (b + a) / a;
double tmp;
if (b <= -1.6e+17) {
tmp = t_0 * (eps / Math.expm1((b * eps)));
} else if (b <= 3.3e-99) {
tmp = t_0 / b;
} else if (b <= 1.55e+227) {
tmp = ((b + a) / b) / a;
} else {
tmp = eps / Math.expm1((a * eps));
}
return tmp;
}
def code(a, b, eps): return (eps * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))
def code(a, b, eps): t_0 = (b + a) / a tmp = 0 if b <= -1.6e+17: tmp = t_0 * (eps / math.expm1((b * eps))) elif b <= 3.3e-99: tmp = t_0 / b elif b <= 1.55e+227: tmp = ((b + a) / b) / a else: tmp = eps / math.expm1((a * eps)) return tmp
function code(a, b, eps) return Float64(Float64(eps * Float64(exp(Float64(Float64(a + b) * eps)) - 1.0)) / Float64(Float64(exp(Float64(a * eps)) - 1.0) * Float64(exp(Float64(b * eps)) - 1.0))) end
function code(a, b, eps) t_0 = Float64(Float64(b + a) / a) tmp = 0.0 if (b <= -1.6e+17) tmp = Float64(t_0 * Float64(eps / expm1(Float64(b * eps)))); elseif (b <= 3.3e-99) tmp = Float64(t_0 / b); elseif (b <= 1.55e+227) tmp = Float64(Float64(Float64(b + a) / b) / a); else tmp = Float64(eps / expm1(Float64(a * eps))); end return tmp end
code[a_, b_, eps_] := N[(N[(eps * N[(N[Exp[N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(a * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Exp[N[(b * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
code[a_, b_, eps_] := Block[{t$95$0 = N[(N[(b + a), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b, -1.6e+17], N[(t$95$0 * N[(eps / N[(Exp[N[(b * eps), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.3e-99], N[(t$95$0 / b), $MachinePrecision], If[LessEqual[b, 1.55e+227], N[(N[(N[(b + a), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], N[(eps / N[(Exp[N[(a * eps), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]]]]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
\begin{array}{l}
t_0 := \frac{b + a}{a}\\
\mathbf{if}\;b \leq -1.6 \cdot 10^{+17}:\\
\;\;\;\;t_0 \cdot \frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\
\mathbf{elif}\;b \leq 3.3 \cdot 10^{-99}:\\
\;\;\;\;\frac{t_0}{b}\\
\mathbf{elif}\;b \leq 1.55 \cdot 10^{+227}:\\
\;\;\;\;\frac{\frac{b + a}{b}}{a}\\
\mathbf{else}:\\
\;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}\\
\end{array}
Results
| Original | 60.3 |
|---|---|
| Target | 15.4 |
| Herbie | 3.5 |
if b < -1.6e17Initial program 43.7
Simplified0.9
Taylor expanded in eps around 0 1.6
Simplified1.6
if -1.6e17 < b < 3.29999999999999986e-99Initial program 64.0
Simplified41.0
Applied egg-rr49.1
Taylor expanded in eps around 0 20.6
Simplified0.1
if 3.29999999999999986e-99 < b < 1.5499999999999999e227Initial program 60.5
Simplified35.8
Taylor expanded in eps around 0 7.5
Simplified3.9
if 1.5499999999999999e227 < b Initial program 49.6
Simplified18.2
Taylor expanded in a around 0 19.1
Final simplification3.5
| Alternative 1 | |
|---|---|
| Error | 4.3 |
| Cost | 6984 |
| Alternative 2 | |
|---|---|
| Error | 9.8 |
| Cost | 712 |
| Alternative 3 | |
|---|---|
| Error | 5.4 |
| Cost | 580 |
| Alternative 4 | |
|---|---|
| Error | 3.7 |
| Cost | 580 |
| Alternative 5 | |
|---|---|
| Error | 14.4 |
| Cost | 324 |
| Alternative 6 | |
|---|---|
| Error | 33.1 |
| Cost | 192 |
herbie shell --seed 2022329
(FPCore (a b eps)
:name "expq3 (problem 3.4.2)"
:precision binary64
:pre (and (< -1.0 eps) (< eps 1.0))
:herbie-target
(/ (+ a b) (* a b))
(/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))