FastMath test2

?

Percentage Accurate: 99.7% → 100.0%
Time: 3.0s
Precision: binary64
Cost: 6720

?

\[\left(d1 \cdot 10 + d1 \cdot d2\right) + d1 \cdot 20 \]
↓
\[\mathsf{fma}\left(d1, 30, d1 \cdot d2\right) \]
(FPCore (d1 d2) :precision binary64 (+ (+ (* d1 10.0) (* d1 d2)) (* d1 20.0)))
↓
(FPCore (d1 d2) :precision binary64 (fma d1 30.0 (* d1 d2)))
double code(double d1, double d2) {
	return ((d1 * 10.0) + (d1 * d2)) + (d1 * 20.0);
}
↓
double code(double d1, double d2) {
	return fma(d1, 30.0, (d1 * d2));
}

function code(d1, d2)
	return Float64(Float64(Float64(d1 * 10.0) + Float64(d1 * d2)) + Float64(d1 * 20.0))
end
↓
function code(d1, d2)
	return fma(d1, 30.0, Float64(d1 * d2))
end

code[d1_, d2_] := N[(N[(N[(d1 * 10.0), $MachinePrecision] + N[(d1 * d2), $MachinePrecision]), $MachinePrecision] + N[(d1 * 20.0), $MachinePrecision]), $MachinePrecision]
↓
code[d1_, d2_] := N[(d1 * 30.0 + N[(d1 * d2), $MachinePrecision]), $MachinePrecision]

\left(d1 \cdot 10 + d1 \cdot d2\right) + d1 \cdot 20
↓
\mathsf{fma}\left(d1, 30, d1 \cdot d2\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:

AlternativeAccuracySpeedup

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

Target

Original99.7%
Target100.0%
Herbie100.0%
\[d1 \cdot \left(30 + d2\right) \]

Derivation?

  1. Initial program 99.8%

    \[\left(d1 \cdot 10 + d1 \cdot d2\right) + d1 \cdot 20 \]

  2. Simplified100.0%

    \[\leadsto \color{blue}{d1 \cdot \left(d2 + 30\right)} \]
    Step-by-step derivation

    [Start]99.8%

    \[ \left(d1 \cdot 10 + d1 \cdot d2\right) + d1 \cdot 20 \]

    +-commutative [=>]99.8%

    \[ \color{blue}{\left(d1 \cdot d2 + d1 \cdot 10\right)} + d1 \cdot 20 \]

    associate-+l+ [=>]99.8%

    \[ \color{blue}{d1 \cdot d2 + \left(d1 \cdot 10 + d1 \cdot 20\right)} \]

    distribute-lft-out [=>]100.0%

    \[ d1 \cdot d2 + \color{blue}{d1 \cdot \left(10 + 20\right)} \]

    distribute-lft-in [<=]100.0%

    \[ \color{blue}{d1 \cdot \left(d2 + \left(10 + 20\right)\right)} \]

    metadata-eval [=>]100.0%

    \[ d1 \cdot \left(d2 + \color{blue}{30}\right) \]

  3. Applied egg-rr100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(d1, 30, d1 \cdot d2\right)} \]
    Step-by-step derivation

    [Start]100.0%

    \[ d1 \cdot \left(d2 + 30\right) \]

    +-commutative [=>]100.0%

    \[ d1 \cdot \color{blue}{\left(30 + d2\right)} \]

    distribute-lft-in [=>]100.0%

    \[ \color{blue}{d1 \cdot 30 + d1 \cdot d2} \]

    *-commutative [<=]100.0%

    \[ d1 \cdot 30 + \color{blue}{d2 \cdot d1} \]

    fma-def [=>]100.0%

    \[ \color{blue}{\mathsf{fma}\left(d1, 30, d2 \cdot d1\right)} \]

    *-commutative [=>]100.0%

    \[ \mathsf{fma}\left(d1, 30, \color{blue}{d1 \cdot d2}\right) \]

  4. Final simplification100.0%

    \[\leadsto \mathsf{fma}\left(d1, 30, d1 \cdot d2\right) \]

Alternatives

Alternative 1
Accuracy100.0%
Cost6720
\[\mathsf{fma}\left(d1, 30, d1 \cdot d2\right) \]
Alternative 2
Accuracy97.7%
Cost456
\[\begin{array}{l} \mathbf{if}\;d2 \leq -30:\\ \;\;\;\;d1 \cdot d2\\ \mathbf{elif}\;d2 \leq 30:\\ \;\;\;\;d1 \cdot 30\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot d2\\ \end{array} \]
Alternative 3
Accuracy100.0%
Cost320
\[d1 \cdot \left(30 + d2\right) \]
Alternative 4
Accuracy50.6%
Cost192
\[d1 \cdot 30 \]

Reproduce?

herbie shell --seed 2023178 
(FPCore (d1 d2)
  :name "FastMath test2"
  :precision binary64

  :herbie-target
  (* d1 (+ 30.0 d2))

  (+ (+ (* d1 10.0) (* d1 d2)) (* d1 20.0)))