Expression 3, p15

?

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Precision: binary64

Cost: 6848

?

\[0 \leq x \land x \leq 2\]

\[x \cdot \left(x \cdot x\right) + x \cdot x \]

↓

\[\mathsf{fma}\left(x, x, x \cdot \left(x \cdot x\right)\right) \]

(FPCore (x) :precision binary64 (+ (* x (* x x)) (* x x)))

↓

(FPCore (x) :precision binary64 (fma x x (* x (* x x))))

double code(double x) {
	return (x * (x * x)) + (x * x);
}

↓

double code(double x) {
	return fma(x, x, (x * (x * x)));
}

function code(x)
	return Float64(Float64(x * Float64(x * x)) + Float64(x * x))
end

↓

function code(x)
	return fma(x, x, Float64(x * Float64(x * x)))
end

code[x_] := N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * x + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x \cdot \left(x \cdot x\right) + x \cdot x

↓

\mathsf{fma}\left(x, x, x \cdot \left(x \cdot x\right)\right)

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 4 alternatives:

Alternative	Accuracy	Speedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

\[\left(\left(1 + x\right) \cdot x\right) \cdot x \]

Derivation?

Initial program 100.0%
\[x \cdot \left(x \cdot x\right) + x \cdot x \]
Taylor expanded in x around 0 100.0%
\[\leadsto \color{blue}{{x}^{2} + {x}^{3}} \]
Simplified100.0%
\[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)} \]
Step-by-step derivation
[Start]100.0
\[ {x}^{2} + {x}^{3} \]
unpow2 [=>]100.0
\[ \color{blue}{x \cdot x} + {x}^{3} \]
fma-udef [<=]100.0
\[ \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)} \]
Applied egg-rr100.0%
\[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(x \cdot x\right) \cdot x}\right) \]
Step-by-step derivation
[Start]100.0
\[ \mathsf{fma}\left(x, x, {x}^{3}\right) \]
unpow3 [=>]100.0
\[ \mathsf{fma}\left(x, x, \color{blue}{\left(x \cdot x\right) \cdot x}\right) \]
Final simplification100.0%
\[\leadsto \mathsf{fma}\left(x, x, x \cdot \left(x \cdot x\right)\right) \]

[Start]100.0	\[ {x}^{2} + {x}^{3} \]
unpow2 [=>]100.0	\[ \color{blue}{x \cdot x} + {x}^{3} \]
fma-udef [<=]100.0	\[ \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)} \]

[Start]100.0	\[ \mathsf{fma}\left(x, x, {x}^{3}\right) \]
unpow3 [=>]100.0	\[ \mathsf{fma}\left(x, x, \color{blue}{\left(x \cdot x\right) \cdot x}\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	576

\[x \cdot x + x \cdot \left(x \cdot x\right) \]

Alternative 2
Accuracy	100.0%
Cost	448

\[x \cdot \left(x + x \cdot x\right) \]

Alternative 3
Accuracy	97.8%
Cost	192

\[x \cdot x \]

Reproduce?

herbie shell --seed 2023160 
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 2.0))

  :herbie-target
  (* (* (+ 1.0 x) x) x)

  (+ (* x (* x x)) (* x x)))