(dist - radius) / sigma

Percentage Accurate: 100.0% → 100.0%
Time: 2.8s
Alternatives: 2
Speedup: 1.0×

Specification

?
\[\left(\left(1 \leq dist \land dist \leq 4\right) \land \left(1 \leq radius \land radius \leq 1\right)\right) \land \left(0.001 \leq sigma \land sigma \leq 0.1\right)\]
\[\begin{array}{l} \\ \frac{dist - radius}{sigma} \end{array} \]
(FPCore (dist radius sigma) :precision binary64 (/ (- dist radius) sigma))
double code(double dist, double radius, double sigma) {
	return (dist - radius) / sigma;
}

real(8) function code(dist, radius, sigma)
    real(8), intent (in) :: dist
    real(8), intent (in) :: radius
    real(8), intent (in) :: sigma
    code = (dist - radius) / sigma
end function

public static double code(double dist, double radius, double sigma) {
	return (dist - radius) / sigma;
}

def code(dist, radius, sigma):
	return (dist - radius) / sigma

function code(dist, radius, sigma)
	return Float64(Float64(dist - radius) / sigma)
end

function tmp = code(dist, radius, sigma)
	tmp = (dist - radius) / sigma;
end

code[dist_, radius_, sigma_] := N[(N[(dist - radius), $MachinePrecision] / sigma), $MachinePrecision]

\begin{array}{l}

\\
\frac{dist - radius}{sigma}
\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: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{dist - radius}{sigma} \end{array} \]
(FPCore (dist radius sigma) :precision binary64 (/ (- dist radius) sigma))
double code(double dist, double radius, double sigma) {
	return (dist - radius) / sigma;
}

real(8) function code(dist, radius, sigma)
    real(8), intent (in) :: dist
    real(8), intent (in) :: radius
    real(8), intent (in) :: sigma
    code = (dist - radius) / sigma
end function

public static double code(double dist, double radius, double sigma) {
	return (dist - radius) / sigma;
}

def code(dist, radius, sigma):
	return (dist - radius) / sigma

function code(dist, radius, sigma)
	return Float64(Float64(dist - radius) / sigma)
end

function tmp = code(dist, radius, sigma)
	tmp = (dist - radius) / sigma;
end

code[dist_, radius_, sigma_] := N[(N[(dist - radius), $MachinePrecision] / sigma), $MachinePrecision]

\begin{array}{l}

\\
\frac{dist - radius}{sigma}
\end{array}

Alternative 1: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{dist - radius}{sigma} \end{array} \]
(FPCore (dist radius sigma) :precision binary64 (/ (- dist radius) sigma))
double code(double dist, double radius, double sigma) {
	return (dist - radius) / sigma;
}

real(8) function code(dist, radius, sigma)
    real(8), intent (in) :: dist
    real(8), intent (in) :: radius
    real(8), intent (in) :: sigma
    code = (dist - radius) / sigma
end function

public static double code(double dist, double radius, double sigma) {
	return (dist - radius) / sigma;
}

def code(dist, radius, sigma):
	return (dist - radius) / sigma

function code(dist, radius, sigma)
	return Float64(Float64(dist - radius) / sigma)
end

function tmp = code(dist, radius, sigma)
	tmp = (dist - radius) / sigma;
end

code[dist_, radius_, sigma_] := N[(N[(dist - radius), $MachinePrecision] / sigma), $MachinePrecision]

\begin{array}{l}

\\
\frac{dist - radius}{sigma}
\end{array}
Derivation

  1. Initial program 100.0%

    \[\frac{dist - radius}{sigma} \]
  2. Add Preprocessing
  3. Add Preprocessing

Alternative 2: 18.7% accurate, 1.3× speedup?

\[\begin{array}{l} \\ \frac{dist}{sigma} \end{array} \]
(FPCore (dist radius sigma) :precision binary64 (/ dist sigma))
double code(double dist, double radius, double sigma) {
	return dist / sigma;
}

real(8) function code(dist, radius, sigma)
    real(8), intent (in) :: dist
    real(8), intent (in) :: radius
    real(8), intent (in) :: sigma
    code = dist / sigma
end function

public static double code(double dist, double radius, double sigma) {
	return dist / sigma;
}

def code(dist, radius, sigma):
	return dist / sigma

function code(dist, radius, sigma)
	return Float64(dist / sigma)
end

function tmp = code(dist, radius, sigma)
	tmp = dist / sigma;
end

code[dist_, radius_, sigma_] := N[(dist / sigma), $MachinePrecision]

\begin{array}{l}

\\
\frac{dist}{sigma}
\end{array}
Derivation

  1. Initial program 100.0%

    \[\frac{dist - radius}{sigma} \]
  2. Add Preprocessing

  3. Taylor expanded in dist around inf

    \[\leadsto \color{blue}{\frac{dist}{sigma}} \]
  4. Step-by-step derivation

    1. lower-/.f6418.6

      \[\leadsto \color{blue}{\frac{dist}{sigma}} \]

  5. Applied rewrites18.6%

    \[\leadsto \color{blue}{\frac{dist}{sigma}} \]
  6. Add Preprocessing

Reproduce

?
herbie shell --seed 1 
(FPCore (dist radius sigma)
  :name "(dist - radius) / sigma"
  :precision binary64
  :pre (and (and (and (<= 1.0 dist) (<= dist 4.0)) (and (<= 1.0 radius) (<= radius 1.0))) (and (<= 0.001 sigma) (<= sigma 0.1)))
  (/ (- dist radius) sigma))