Octave 3.8, jcobi/1

?

Percentage Accurate: 74.4% → 99.9%
Time: 17.5s
Precision: binary64
Cost: 8388

?

\[\alpha > -1 \land \beta > -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
↓
\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]
(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))
↓
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99995)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/ (+ t_0 1.0) 2.0))))
double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}
↓
double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99995) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end
↓
function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99995)
		tmp = Float64(fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(Float64(beta + 2.0) / alpha), Float64(Float64(beta + Float64(beta + 2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]
↓
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99995], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\
\mathbf{if}\;t_0 \leq -0.99995:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0 + 1}{2}\\


\end{array}

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 8 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

Derivation?

  1. Split input into 2 regimes

  2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999950000000000006

    1. Initial program 6.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

    2. Simplified6.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
      Step-by-step derivation

      [Start]6.0%

      \[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]

      +-commutative [=>]6.0%

      \[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

    3. Taylor expanded in alpha around -inf 95.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}}{2} \]

    4. Simplified100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{2 + \beta}{\alpha}, \frac{\beta + \left(2 + \beta\right)}{\alpha}\right)}}{2} \]
      Step-by-step derivation

      [Start]95.6%

      \[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]

      +-commutative [=>]95.6%

      \[ \frac{\color{blue}{-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + -1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}}}{2} \]

      mul-1-neg [=>]95.6%

      \[ \frac{-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} + \color{blue}{\left(-\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}\right)}}{2} \]

      unsub-neg [=>]95.6%

      \[ \frac{\color{blue}{-1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}} - \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}}}{2} \]

    if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

    1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification99.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy99.9%
Cost8388
\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99995:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]
Alternative 2
Accuracy99.6%
Cost1476
\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]
Alternative 3
Accuracy86.6%
Cost708
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
Alternative 4
Accuracy91.9%
Cost708
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
Alternative 5
Accuracy71.3%
Cost580
\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 6
Accuracy71.6%
Cost580
\[\begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]
Alternative 7
Accuracy50.9%
Cost452
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.5 \cdot 10^{+51}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
Alternative 8
Accuracy36.6%
Cost64
\[1 \]

Reproduce?

herbie shell --seed 2023263 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))