Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B

?

Percentage Accurate: 99.8% → 99.8%

Time: 4.8s

Precision: binary64

Cost: 13120

?

\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]

↓

\[0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

↓

(FPCore (x y z) :precision binary64 (* 0.5 (fma y (sqrt z) x)))

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

↓

double code(double x, double y, double z) {
	return 0.5 * fma(y, sqrt(z), x);
}

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

↓

function code(x, y, z)
	return Float64(0.5 * fma(y, sqrt(z), x))
end

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)

↓

0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)

Local Percentage Accuracy vs ?

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 alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

Initial program 99.8%
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]

Simplified99.9%

\[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]

Step-by-step derivation

[Start]99.8%	\[ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
metadata-eval [=>]99.8%	\[ \color{blue}{0.5} \cdot \left(x + y \cdot \sqrt{z}\right) \]
+-commutative [=>]99.8%	\[ 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z} + x\right)} \]
fma-def [=>]99.9%	\[ 0.5 \cdot \color{blue}{\mathsf{fma}\left(y, \sqrt{z}, x\right)} \]

Final simplification99.9%
\[\leadsto 0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13120

\[0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]

Alternative 2
Accuracy	74.6%
Cost	20041

\[\begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-67} \lor \neg \left(t_0 \leq 5 \cdot 10^{-53}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	6848

\[0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \]

Alternative 4
Accuracy	51.2%
Cost	192

\[0.5 \cdot x \]

Reproduce?

herbie shell --seed 2023171 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))