bug366, discussion (missed optimization)

Percentage Accurate: 53.7% → 99.5%
Time: 3.6s
Alternatives: 2
Speedup: 24.0×

Specification

?
\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]
(FPCore (a b) :precision binary64 (sqrt (- (* a a) (* b b))))
double code(double a, double b) {
	return sqrt(((a * a) - (b * b)));
}

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(((a * a) - (b * b)))
end function

public static double code(double a, double b) {
	return Math.sqrt(((a * a) - (b * b)));
}

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

function code(a, b)
	return sqrt(Float64(Float64(a * a) - Float64(b * b)))
end

function tmp = code(a, b)
	tmp = sqrt(((a * a) - (b * b)));
end

code[a_, b_] := N[Sqrt[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\sqrt{a \cdot a - b \cdot b}
\end{array}

Sampling outcomes in binary64 precision:

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.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
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.

Initial Program: 53.7% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]
(FPCore (a b) :precision binary64 (sqrt (- (* a a) (* b b))))
double code(double a, double b) {
	return sqrt(((a * a) - (b * b)));
}

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(((a * a) - (b * b)))
end function

public static double code(double a, double b) {
	return Math.sqrt(((a * a) - (b * b)));
}

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

function code(a, b)
	return sqrt(Float64(Float64(a * a) - Float64(b * b)))
end

function tmp = code(a, b)
	tmp = sqrt(((a * a) - (b * b)));
end

code[a_, b_] := N[Sqrt[N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\sqrt{a \cdot a - b \cdot b}
\end{array}

Alternative 1: 99.5% accurate, 1.0× speedup?

\[\begin{array}{l} a_m = \left|a\right| \\ \mathsf{fma}\left(\frac{b}{a\_m} \cdot b, -0.5, a\_m\right) \end{array} \]
a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 (fma (* (/ b a_m) b) -0.5 a_m))
a_m = fabs(a);
double code(double a_m, double b) {
	return fma(((b / a_m) * b), -0.5, a_m);
}

a_m = abs(a)
function code(a_m, b)
	return fma(Float64(Float64(b / a_m) * b), -0.5, a_m)
end

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(N[(N[(b / a$95$m), $MachinePrecision] * b), $MachinePrecision] * -0.5 + a$95$m), $MachinePrecision]

\begin{array}{l}
a_m = \left|a\right|

\\
\mathsf{fma}\left(\frac{b}{a\_m} \cdot b, -0.5, a\_m\right)
\end{array}
Derivation

  1. Initial program 50.7%

    \[\sqrt{a \cdot a - b \cdot b} \]
  2. Add Preprocessing

  3. Taylor expanded in b around 0

    \[\leadsto \color{blue}{a + \frac{-1}{2} \cdot \frac{{b}^{2}}{a}} \]
  4. Step-by-step derivation

    1. +-commutativeN/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{b}^{2}}{a} + a} \]

    2. *-commutativeN/A

      \[\leadsto \color{blue}{\frac{{b}^{2}}{a} \cdot \frac{-1}{2}} + a \]

    3. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{b}^{2}}{a}, \frac{-1}{2}, a\right)} \]

    4. lower-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{b}^{2}}{a}}, \frac{-1}{2}, a\right) \]

    5. unpow2N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{b \cdot b}}{a}, \frac{-1}{2}, a\right) \]

    6. lower-*.f6448.2

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{b \cdot b}}{a}, -0.5, a\right) \]

  5. Applied rewrites48.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b \cdot b}{a}, -0.5, a\right)} \]
  6. Step-by-step derivation

    1. Applied rewrites50.5%

      \[\leadsto \mathsf{fma}\left(\frac{b}{a} \cdot b, -0.5, a\right) \]
    2. Add Preprocessing

    Alternative 2: 99.1% accurate, 24.0× speedup?

    \[\begin{array}{l} a_m = \left|a\right| \\ a\_m \end{array} \]
    a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 a_m)
    a_m = fabs(a);
double code(double a_m, double b) {
	return a_m;
}

    a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = a_m
end function

    a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return a_m;
}

    a_m = math.fabs(a)
def code(a_m, b):
	return a_m

    a_m = abs(a)
function code(a_m, b)
	return a_m
end

    a_m = abs(a);
function tmp = code(a_m, b)
	tmp = a_m;
end

    a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := a$95$m

    \begin{array}{l}
a_m = \left|a\right|

\\
a\_m
\end{array}
    Derivation

    1. Initial program 50.7%

      \[\sqrt{a \cdot a - b \cdot b} \]
    2. Add Preprocessing

    3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{-1 \cdot a} \]
    4. Step-by-step derivation

      1. mul-1-negN/A

        \[\leadsto \color{blue}{\mathsf{neg}\left(a\right)} \]

      2. lower-neg.f6449.7

        \[\leadsto \color{blue}{-a} \]

    5. Applied rewrites49.7%

      \[\leadsto \color{blue}{-a} \]
    6. Step-by-step derivation

      1. Applied rewrites50.0%

        \[\leadsto \color{blue}{a} \]
      2. Add Preprocessing

      Developer Target 1: 99.2% accurate, 0.6× speedup?

      \[\begin{array}{l} \\ \sqrt{\left|a\right| + \left|b\right|} \cdot \sqrt{\left|a\right| - \left|b\right|} \end{array} \]
      (FPCore (a b)
 :precision binary64
 (* (sqrt (+ (fabs a) (fabs b))) (sqrt (- (fabs a) (fabs b)))))
      double code(double a, double b) {
	return sqrt((fabs(a) + fabs(b))) * sqrt((fabs(a) - fabs(b)));
}

      real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt((abs(a) + abs(b))) * sqrt((abs(a) - abs(b)))
end function

      public static double code(double a, double b) {
	return Math.sqrt((Math.abs(a) + Math.abs(b))) * Math.sqrt((Math.abs(a) - Math.abs(b)));
}

      def code(a, b):
	return math.sqrt((math.fabs(a) + math.fabs(b))) * math.sqrt((math.fabs(a) - math.fabs(b)))

      function code(a, b)
	return Float64(sqrt(Float64(abs(a) + abs(b))) * sqrt(Float64(abs(a) - abs(b))))
end

      function tmp = code(a, b)
	tmp = sqrt((abs(a) + abs(b))) * sqrt((abs(a) - abs(b)));
end

      code[a_, b_] := N[(N[Sqrt[N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[a], $MachinePrecision] - N[Abs[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

      \begin{array}{l}

\\
\sqrt{\left|a\right| + \left|b\right|} \cdot \sqrt{\left|a\right| - \left|b\right|}
\end{array}

      Reproduce

      ?
      herbie shell --seed 2024305 
(FPCore (a b)
  :name "bug366, discussion (missed optimization)"
  :precision binary64

  :alt
  (! :herbie-platform default (let* ((fa (fabs a)) (fb (fabs b))) (* (sqrt (+ fa fb)) (sqrt (- fa fb)))))

  (sqrt (- (* a a) (* b b))))