Average Error: 0.4 → 0.2
Time: 6.7s
Precision: binary64
\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
↓
\[\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(d \cdot d - b \cdot b, \frac{1}{d - b}, c + a\right)}, \sqrt{a + \left(c + \left(d + b\right)\right)}, e\right) \]
(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))
↓
(FPCore (a b c d e)
 :precision binary64
 (fma
  (sqrt (fma (- (* d d) (* b b)) (/ 1.0 (- d b)) (+ c a)))
  (sqrt (+ a (+ c (+ d b))))
  e))
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 fma(sqrt(fma(((d * d) - (b * b)), (1.0 / (d - b)), (c + a))), sqrt((a + (c + (d + b)))), e);
}

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 fma(sqrt(fma(Float64(Float64(d * d) - Float64(b * b)), Float64(1.0 / Float64(d - b)), Float64(c + a))), sqrt(Float64(a + Float64(c + Float64(d + b)))), e)
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[(N[Sqrt[N[(N[(N[(d * d), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(d - b), $MachinePrecision]), $MachinePrecision] + N[(c + a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a + N[(c + N[(d + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + e), $MachinePrecision]

\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(d \cdot d - b \cdot b, \frac{1}{d - b}, c + a\right)}, \sqrt{a + \left(c + \left(d + b\right)\right)}, e\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
Herbie0.2
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e \]

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-rr0.3

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

  4. Applied egg-rr0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022162 
(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))