fabs fraction 1

Percentage Accurate: 91.5% → 99.9%
Time: 4.9s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \end{array} \]
(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
\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 3 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: 91.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \end{array} \]
(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
\end{array}

Alternative 1: 99.9% accurate, 0.5× speedup?

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.6 \cdot 10^{-22}:\\ \;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\ \end{array} \end{array} \]
y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 8.6e-22)
   (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x))))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))
y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8.6e-22) {
		tmp = fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 8.6e-22)
		tmp = abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 8.6e-22], N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

\begin{array}{l}
y_m = \left|y\right|

\\
\begin{array}{l}
\mathbf{if}\;y\_m \leq 8.6 \cdot 10^{-22}:\\
\;\;\;\;\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y\_m}, \frac{-4 - x}{y\_m}\right)\right|\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if y < 8.60000000000000075e-22

    1. Initial program 92.4%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

    3. Simplified98.9%

      \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
    4. Add Preprocessing

    if 8.60000000000000075e-22 < y

    1. Initial program 96.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
    2. Step-by-step derivation

      1. fabs-sub96.6%

        \[\leadsto \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]

      2. associate-*l/90.6%

        \[\leadsto \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]

      3. associate-*r/99.9%

        \[\leadsto \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]

      4. fma-neg99.9%

        \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]

      5. distribute-neg-frac99.9%

        \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]

      6. +-commutative99.9%

        \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-\color{blue}{\left(4 + x\right)}}{y}\right)\right| \]

      7. distribute-neg-in99.9%

        \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) + \left(-x\right)}}{y}\right)\right| \]

      8. unsub-neg99.9%

        \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(-4\right) - x}}{y}\right)\right| \]

      9. metadata-eval99.9%

        \[\leadsto \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

    3. Simplified99.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|} \]
    4. Add Preprocessing
  3. Recombined 2 regimes into one program.

  4. Final simplification99.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-22}:\\ \;\;\;\;\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
  5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.5× speedup?

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right| \end{array} \]
y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (fabs (* (/ -1.0 y_m) (fma x z (- -4.0 x)))))
y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs(((-1.0 / y_m) * fma(x, z, (-4.0 - x))));
}

y_m = abs(y)
function code(x, y_m, z)
	return abs(Float64(Float64(-1.0 / y_m) * fma(x, z, Float64(-4.0 - x))))
end

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(N[(-1.0 / y$95$m), $MachinePrecision] * N[(x * z + N[(-4.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}
y_m = \left|y\right|

\\
\left|\frac{-1}{y\_m} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|
\end{array}
Derivation

  1. Initial program 93.4%

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

  3. Simplified97.0%

    \[\leadsto \color{blue}{\left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right|} \]
  4. Add Preprocessing

  5. Final simplification97.0%

    \[\leadsto \left|\frac{-1}{y} \cdot \mathsf{fma}\left(x, z, -4 - x\right)\right| \]
  6. Add Preprocessing

Alternative 3: 91.5% accurate, 1.0× speedup?

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right| \end{array} \]
y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m)))))
y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs((((x + 4.0) / y_m) - (z * (x / y_m))));
}

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y_m) - (z * (x / y_m))))
end function

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	return Math.abs((((x + 4.0) / y_m) - (z * (x / y_m))));
}

y_m = math.fabs(y)
def code(x, y_m, z):
	return math.fabs((((x + 4.0) / y_m) - (z * (x / y_m))))

y_m = abs(y)
function code(x, y_m, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m))))
end

y_m = abs(y);
function tmp = code(x, y_m, z)
	tmp = abs((((x + 4.0) / y_m) - (z * (x / y_m))));
end

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\begin{array}{l}
y_m = \left|y\right|

\\
\left|\frac{x + 4}{y\_m} - z \cdot \frac{x}{y\_m}\right|
\end{array}
Derivation

  1. Initial program 93.4%

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
  2. Add Preprocessing

  3. Final simplification93.4%

    \[\leadsto \left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \]
  4. Add Preprocessing

Reproduce

?
herbie shell --seed 2024043 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))