Average Error: 0.4 → 0.0
Time: 4.9s
Precision: binary64
\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
↓
\[\log \left(\left(1 + \mathsf{fma}\left(e^{d}, e^{a} \cdot e^{b}, -1\right)\right) \cdot \left(e^{e} \cdot e^{c}\right)\right) \]
(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))
↓
(FPCore (a b c d e)
 :precision binary64
 (log (* (+ 1.0 (fma (exp d) (* (exp a) (exp b)) -1.0)) (* (exp e) (exp c)))))
double code(double a, double b, double c, double d, double e) {
	return (((e + d) + c) + b) + a;
}
↓
double code(double a, double b, double c, double d, double e) {
	return log(((1.0 + fma(exp(d), (exp(a) * exp(b)), -1.0)) * (exp(e) * exp(c))));
}

function code(a, b, c, d, e)
	return Float64(Float64(Float64(Float64(e + d) + c) + b) + a)
end
↓
function code(a, b, c, d, e)
	return log(Float64(Float64(1.0 + fma(exp(d), Float64(exp(a) * exp(b)), -1.0)) * Float64(exp(e) * exp(c))))
end

code[a_, b_, c_, d_, e_] := N[(N[(N[(N[(e + d), $MachinePrecision] + c), $MachinePrecision] + b), $MachinePrecision] + a), $MachinePrecision]
↓
code[a_, b_, c_, d_, e_] := N[Log[N[(N[(1.0 + N[(N[Exp[d], $MachinePrecision] * N[(N[Exp[a], $MachinePrecision] * N[Exp[b], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[e], $MachinePrecision] * N[Exp[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(\left(1 + \mathsf{fma}\left(e^{d}, e^{a} \cdot e^{b}, -1\right)\right) \cdot \left(e^{e} \cdot e^{c}\right)\right)

Error

Target

Original0.4
Target0.2
Herbie0.0
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e \]

Derivation

  1. Initial program 0.4

    \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]

  2. Simplified0.2

    \[\leadsto \color{blue}{e + \left(c + \left(a + \left(d + b\right)\right)\right)} \]

  3. Applied egg-rr0.1

    \[\leadsto \color{blue}{\log \left(\left(1 + \mathsf{expm1}\left(d + \left(b + a\right)\right)\right) \cdot \left(e^{e} \cdot e^{c}\right)\right)} \]

  4. Applied egg-rr0.1

    \[\leadsto \log \left(\left(1 + \color{blue}{\mathsf{fma}\left(e^{d}, e^{b + a}, -1\right)}\right) \cdot \left(e^{e} \cdot e^{c}\right)\right) \]

  5. Applied egg-rr0.0

    \[\leadsto \log \left(\left(1 + \mathsf{fma}\left(e^{d}, \color{blue}{e^{a} \cdot e^{b}}, -1\right)\right) \cdot \left(e^{e} \cdot e^{c}\right)\right) \]

  6. Final simplification0.0

    \[\leadsto \log \left(\left(1 + \mathsf{fma}\left(e^{d}, e^{a} \cdot e^{b}, -1\right)\right) \cdot \left(e^{e} \cdot e^{c}\right)\right) \]

Reproduce

herbie shell --seed 2022210 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 a) (<= a 2.0)) (<= 2.0 b)) (<= b 4.0)) (<= 4.0 c)) (<= c 8.0)) (<= 8.0 d)) (<= d 16.0)) (<= 16.0 e)) (<= e 32.0))

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))