Expression, p14

Average Error: 0.0 → 0.0

Time: 11.6s

Precision: binary64

Cost: 6848

\[\left(\left(\left(56789 \leq a \land a \leq 98765\right) \land \left(0 \leq b \land b \leq 1\right)\right) \land \left(0 \leq c \land c \leq 0.0016773\right)\right) \land \left(0 \leq d \land d \leq 0.0016773\right)\]

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c d) :precision binary64 (fma (+ c d) a (* a b)))

C

double code(double a, double b, double c, double d) {
	return a * ((b + c) + d);
}

↓

double code(double a, double b, double c, double d) {
	return fma((c + d), a, (a * b));
}

Julia

function code(a, b, c, d)
	return Float64(a * Float64(Float64(b + c) + d))
end

↓

function code(a, b, c, d)
	return fma(Float64(c + d), a, Float64(a * b))
end

Wolfram

code[a_, b_, c_, d_] := N[(a * N[(N[(b + c), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_, d_] := N[(N[(c + d), $MachinePrecision] * a + N[(a * b), $MachinePrecision]), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	30.5
Cost	1380

\[\begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-297}:\\ \;\;\;\;c \cdot a\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-276}:\\ \;\;\;\;a \cdot d\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-257}:\\ \;\;\;\;c \cdot a\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-239}:\\ \;\;\;\;a \cdot d\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-190}:\\ \;\;\;\;c \cdot a\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-174}:\\ \;\;\;\;a \cdot d\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-134}:\\ \;\;\;\;c \cdot a\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-97}:\\ \;\;\;\;a \cdot b\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-89}:\\ \;\;\;\;c \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 2
Error	13.9
Cost	716

\[\begin{array}{l} t_0 := a \cdot \left(c + b\right)\\ \mathbf{if}\;d \leq 9.5 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-61}:\\ \;\;\;\;a \cdot d\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot d\\ \end{array} \]

Alternative 3
Error	9.6
Cost	716

\[\begin{array}{l} t_0 := \left(d + b\right) \cdot a\\ t_1 := a \cdot \left(c + d\right)\\ \mathbf{if}\;b \leq 3 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.0
Cost	576

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

Alternative 5
Error	9.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-106}:\\ \;\;\;\;a \cdot \left(c + d\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(c + b\right)\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	30.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-106}:\\ \;\;\;\;a \cdot d\\ \mathbf{else}:\\ \;\;\;\;a \cdot b\\ \end{array} \]

Alternative 8
Error	41.4
Cost	192

\[a \cdot b \]

Reproduce

herbie shell --seed 2023010 
(FPCore (a b c d)
  :name "Expression, p14"
  :precision binary64
  :pre (and (and (and (and (<= 56789.0 a) (<= a 98765.0)) (and (<= 0.0 b) (<= b 1.0))) (and (<= 0.0 c) (<= c 0.0016773))) (and (<= 0.0 d) (<= d 0.0016773)))

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

  (* a (+ (+ b c) d)))