Octave 3.8, jcobi/1

Average Error: 24.72% → 0.26%

Time: 13.6s

Precision: binary64

Cost: 15812

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta - \alpha}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\frac{2}{\alpha} - \mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {t_1}^{3}}{{t_1}^{2} + \left(1 + \frac{\alpha - \beta}{t_0}\right)}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ (- beta alpha) t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999)
     (/
      (-
       (/ 2.0 alpha)
       (fma
        beta
        (- (/ 6.0 (* alpha alpha)) (/ 2.0 alpha))
        (/ 4.0 (* alpha alpha))))
      2.0)
     (/
      (/
       (+ 1.0 (pow t_1 3.0))
       (+ (pow t_1 2.0) (+ 1.0 (/ (- alpha beta) t_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 t_0 = beta + (alpha + 2.0);
	double t_1 = (beta - alpha) / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999) {
		tmp = ((2.0 / alpha) - fma(beta, ((6.0 / (alpha * alpha)) - (2.0 / alpha)), (4.0 / (alpha * alpha)))) / 2.0;
	} else {
		tmp = ((1.0 + pow(t_1, 3.0)) / (pow(t_1, 2.0) + (1.0 + ((alpha - beta) / t_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)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(Float64(beta - alpha) / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999999)
		tmp = Float64(Float64(Float64(2.0 / alpha) - fma(beta, Float64(Float64(6.0 / Float64(alpha * alpha)) - Float64(2.0 / alpha)), Float64(4.0 / Float64(alpha * alpha)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + (t_1 ^ 3.0)) / Float64((t_1 ^ 2.0) + Float64(1.0 + Float64(Float64(alpha - beta) / t_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_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999], N[(N[(N[(2.0 / alpha), $MachinePrecision] - N[(beta * N[(N[(6.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(1.0 + N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999998999999999971

   1. Initial program 93.25

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

   2. Simplified 93.25

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

      [Start] 93.25	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 93.25	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 4.6

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

   4. Simplified 4.6

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

      [Start] 4.6	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + -1 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}}{2} \]
      distribute-lft-out [=>] 4.6	\[ \frac{\color{blue}{-1 \cdot \left(\frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}}{2} \]
      mul-1-neg [=>] 4.6	\[ \frac{-1 \cdot \left(\frac{\color{blue}{\left(-\beta\right)} - \left(\beta + 2\right)}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 4.6	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \color{blue}{\left(2 + \beta\right)}}{\alpha} + \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 4.6	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\color{blue}{\left(2 + \beta\right)}}^{2} + \beta \cdot \left(\beta + 2\right)}{{\alpha}^{2}}\right)}{2} \]
      +-commutative [=>] 4.6	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \color{blue}{\left(2 + \beta\right)}}{{\alpha}^{2}}\right)}{2} \]
      unpow2 [=>] 4.6	\[ \frac{-1 \cdot \left(\frac{\left(-\beta\right) - \left(2 + \beta\right)}{\alpha} + \frac{{\left(2 + \beta\right)}^{2} + \beta \cdot \left(2 + \beta\right)}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   5. Taylor expanded in beta around 0 0.38

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(\beta \cdot \left(6 \cdot \frac{1}{{\alpha}^{2}} - 2 \cdot \frac{1}{\alpha}\right) + 4 \cdot \frac{1}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}}{2} \]

   6. Simplified 0.38

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{4}{\alpha \cdot \alpha}\right) - \frac{2}{\alpha}\right)}}{2} \]
      Proof

      [Start] 0.38	\[ \frac{-1 \cdot \left(\left(\beta \cdot \left(6 \cdot \frac{1}{{\alpha}^{2}} - 2 \cdot \frac{1}{\alpha}\right) + 4 \cdot \frac{1}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      fma-def [=>] 0.38	\[ \frac{-1 \cdot \left(\color{blue}{\mathsf{fma}\left(\beta, 6 \cdot \frac{1}{{\alpha}^{2}} - 2 \cdot \frac{1}{\alpha}, 4 \cdot \frac{1}{{\alpha}^{2}}\right)} - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      associate-*r/ [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \color{blue}{\frac{6 \cdot 1}{{\alpha}^{2}}} - 2 \cdot \frac{1}{\alpha}, 4 \cdot \frac{1}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      metadata-eval [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{\color{blue}{6}}{{\alpha}^{2}} - 2 \cdot \frac{1}{\alpha}, 4 \cdot \frac{1}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      unpow2 [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\color{blue}{\alpha \cdot \alpha}} - 2 \cdot \frac{1}{\alpha}, 4 \cdot \frac{1}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      associate-*r/ [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \color{blue}{\frac{2 \cdot 1}{\alpha}}, 4 \cdot \frac{1}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      metadata-eval [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{\color{blue}{2}}{\alpha}, 4 \cdot \frac{1}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      associate-*r/ [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      metadata-eval [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{\color{blue}{4}}{{\alpha}^{2}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      unpow2 [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{4}{\color{blue}{\alpha \cdot \alpha}}\right) - 2 \cdot \frac{1}{\alpha}\right)}{2} \]
      associate-*r/ [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{4}{\alpha \cdot \alpha}\right) - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)}{2} \]
      metadata-eval [=>] 0.38	\[ \frac{-1 \cdot \left(\mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{4}{\alpha \cdot \alpha}\right) - \frac{\color{blue}{2}}{\alpha}\right)}{2} \]

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.16

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

   2. Simplified 0.16

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

      [Start] 0.16	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.16	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.21

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

4. Final simplification 0.26

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\frac{2}{\alpha} - \mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}{{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(1 + \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.27%
Cost	8388

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999:\\ \;\;\;\;\frac{\frac{2}{\alpha} - \mathsf{fma}\left(\beta, \frac{6}{\alpha \cdot \alpha} - \frac{2}{\alpha}, \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\beta + \alpha}, \left(\beta + \alpha\right) \cdot t_0, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.29%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 3
Error	0.28%
Cost	1860

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha} + \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 4
Error	0.34%
Cost	1476

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.999999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \]

Alternative 5
Error	11.78%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 11500:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	6.75%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	30.52%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.25:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	31%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	28.3%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 190:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	50.23%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023121 
(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))