Octave 3.8, jcobi/1

Average Error: 26.09% → 0.44%

Time: 10.4s

Precision: binary64

Cost: 7876

\[\alpha > -1 \land \beta > -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

↓

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99996)
   (/ (+ beta 1.0) alpha)
   (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99996) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

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)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99996)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

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_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99996], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99995999999999996

   1. Initial program 92.64

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

   2. Simplified 92.64

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

      [Start] 92.64	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 92.64	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 1.33

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]

   4. Taylor expanded in alpha around 0 1.33

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   5. Simplified 1.33

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]
      Proof

      [Start] 1.33	\[ 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]
      +-commutative [=>] 1.33	\[ 0.5 \cdot \frac{\color{blue}{2 \cdot \beta + 2}}{\alpha} \]
      fma-udef [<=] 1.33	\[ 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \beta, 2\right)}}{\alpha} \]
      metadata-eval [<=] 1.33	\[ \color{blue}{\frac{1}{2}} \cdot \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha} \]
      associate-/r/ [<=] 1.37	\[ \color{blue}{\frac{1}{\frac{2}{\frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}}} \]
      *-lft-identity [<=] 1.37	\[ \frac{1}{\frac{2}{\color{blue}{1 \cdot \frac{\mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}}} \]
      associate-*r/ [=>] 1.37	\[ \frac{1}{\frac{2}{\color{blue}{\frac{1 \cdot \mathsf{fma}\left(2, \beta, 2\right)}{\alpha}}}} \]
      associate-*l/ [<=] 1.43	\[ \frac{1}{\frac{2}{\color{blue}{\frac{1}{\alpha} \cdot \mathsf{fma}\left(2, \beta, 2\right)}}} \]
      fma-udef [=>] 1.43	\[ \frac{1}{\frac{2}{\frac{1}{\alpha} \cdot \color{blue}{\left(2 \cdot \beta + 2\right)}}} \]
      distribute-rgt-out [<=] 1.43	\[ \frac{1}{\frac{2}{\color{blue}{\left(2 \cdot \beta\right) \cdot \frac{1}{\alpha} + 2 \cdot \frac{1}{\alpha}}}} \]
      associate-*l* [=>] 1.43	\[ \frac{1}{\frac{2}{\color{blue}{2 \cdot \left(\beta \cdot \frac{1}{\alpha}\right)} + 2 \cdot \frac{1}{\alpha}}} \]
      distribute-lft-out [=>] 1.43	\[ \frac{1}{\frac{2}{\color{blue}{2 \cdot \left(\beta \cdot \frac{1}{\alpha} + \frac{1}{\alpha}\right)}}} \]
      distribute-lft1-in [=>] 1.43	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\left(\left(\beta + 1\right) \cdot \frac{1}{\alpha}\right)}}} \]
      /-rgt-identity [<=] 1.43	\[ \frac{1}{\frac{2}{2 \cdot \left(\color{blue}{\frac{\beta + 1}{1}} \cdot \frac{1}{\alpha}\right)}} \]
      associate-*r/ [=>] 1.37	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\frac{\frac{\beta + 1}{1} \cdot 1}{\alpha}}}} \]
      associate-*l/ [<=] 1.37	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\left(\frac{\frac{\beta + 1}{1}}{\alpha} \cdot 1\right)}}} \]
      *-rgt-identity [=>] 1.37	\[ \frac{1}{\frac{2}{2 \cdot \color{blue}{\frac{\frac{\beta + 1}{1}}{\alpha}}}} \]
      associate-/r* [=>] 1.37	\[ \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\frac{\frac{\beta + 1}{1}}{\alpha}}}} \]
      metadata-eval [=>] 1.37	\[ \frac{1}{\frac{\color{blue}{1}}{\frac{\frac{\beta + 1}{1}}{\alpha}}} \]
      associate-/l* [<=] 1.33	\[ \color{blue}{\frac{1 \cdot \frac{\frac{\beta + 1}{1}}{\alpha}}{1}} \]

   if -0.99995999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.09

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

   2. Simplified 0.09

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

      [Start] 0.09	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.09	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.09

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.44

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.44%
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 2
Error	0.44%
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99996:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 3
Error	34.52%
Cost	1116

\[\begin{array}{l} \mathbf{if}\;\beta \leq -2.5 \cdot 10^{-201}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -1.22 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 32:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+43}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 4.1 \cdot 10^{+113}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+191}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 1.3 \cdot 10^{+203}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	25.52%
Cost	844

\[\begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -1.1 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.8:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 5
Error	25.92%
Cost	716

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2.1 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 54:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 6
Error	6.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 7
Error	31.57%
Cost	588

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -1.5 \cdot 10^{-130}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -1.1 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 53:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 8
Error	31.03%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 53:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 9
Error	51.05%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023102 
(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))