Average Error: 0.4 → 0.3
Time: 3.7s
Precision: binary64
\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
↓
\[e + \mathsf{fma}\left(\sqrt{c}, \sqrt{c}, \mathsf{fma}\left(\sqrt[3]{a}, \sqrt[3]{a \cdot a}, d + b\right)\right) \]
(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))
↓
(FPCore (a b c d e)
 :precision binary64
 (+ e (fma (sqrt c) (sqrt c) (fma (cbrt a) (cbrt (* a a)) (+ d b)))))
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 e + fma(sqrt(c), sqrt(c), fma(cbrt(a), cbrt((a * a)), (d + b)));
}

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 Float64(e + fma(sqrt(c), sqrt(c), fma(cbrt(a), cbrt(Float64(a * a)), Float64(d + b))))
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[(e + N[(N[Sqrt[c], $MachinePrecision] * N[Sqrt[c], $MachinePrecision] + N[(N[Power[a, 1/3], $MachinePrecision] * N[Power[N[(a * a), $MachinePrecision], 1/3], $MachinePrecision] + N[(d + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
e + \mathsf{fma}\left(\sqrt{c}, \sqrt{c}, \mathsf{fma}\left(\sqrt[3]{a}, \sqrt[3]{a \cdot a}, d + b\right)\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original0.4
Target0.2
Herbie0.3
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e \]

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-rr0.2

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

  4. Applied egg-rr0.3

    \[\leadsto e + \mathsf{fma}\left(\sqrt{c}, \sqrt{c}, \color{blue}{\mathsf{fma}\left(\sqrt[3]{a}, {\left(\sqrt[3]{a}\right)}^{2}, d + b\right)}\right) \]

  5. Applied egg-rr0.3

    \[\leadsto e + \mathsf{fma}\left(\sqrt{c}, \sqrt{c}, \mathsf{fma}\left(\sqrt[3]{a}, \color{blue}{\sqrt[3]{a \cdot a}}, d + b\right)\right) \]

  6. Final simplification0.3

    \[\leadsto e + \mathsf{fma}\left(\sqrt{c}, \sqrt{c}, \mathsf{fma}\left(\sqrt[3]{a}, \sqrt[3]{a \cdot a}, d + b\right)\right) \]

Reproduce

herbie shell --seed 2022166 
(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))