Expression 1, p15

Average Error: 0.4 → 0.2

Time: 5.9s

Precision: binary64

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]

Math

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

↓

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))

↓

(FPCore (a b c d e)
 :precision binary64
 (+
  (fma
   (cbrt (* b b))
   (* (sqrt (cbrt (* b (cbrt b)))) (sqrt (pow (cbrt (cbrt b)) 2.0)))
   (+ a d))
  (+ e c)))

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 fma(cbrt((b * b)), (sqrt(cbrt((b * cbrt(b)))) * sqrt(pow(cbrt(cbrt(b)), 2.0))), (a + d)) + (e + c);
}

Julia

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(fma(cbrt(Float64(b * b)), Float64(sqrt(cbrt(Float64(b * cbrt(b)))) * sqrt((cbrt(cbrt(b)) ^ 2.0))), Float64(a + d)) + Float64(e + c))
end

Wolfram

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[(N[Power[N[(b * b), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Sqrt[N[Power[N[(b * N[Power[b, 1/3], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Power[N[Power[N[Power[b, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(a + d), $MachinePrecision]), $MachinePrecision] + N[(e + c), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original	0.4
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.4

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

2. Simplified 0.2

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt[3]{b \cdot b}, \sqrt[3]{b}, a + d\right)} + \left(e + c\right)\right) \cdot 1 \]

5. Applied egg-rr 0.2

   \[\leadsto \left(\mathsf{fma}\left(\sqrt[3]{b \cdot b}, \color{blue}{\sqrt{\sqrt[3]{b \cdot \sqrt[3]{b}}} \cdot \sqrt{{\left(\sqrt[3]{\sqrt[3]{b}}\right)}^{2}}}, a + d\right) + \left(e + c\right)\right) \cdot 1 \]

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022159 
(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))