Octave 3.8, jcobi/1

Average Accuracy: 74.2% → 99.3%

Time: 20.0s

Precision: binary64

Cost: 27652

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

Math

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

↓

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

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ beta (+ alpha 2.0)))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
     (+ (/ beta alpha) (/ 1.0 alpha))
     (/ (fma (cbrt (pow t_0 2.0)) (cbrt t_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 t_0 = (beta - alpha) / (beta + (alpha + 2.0));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = (beta / alpha) + (1.0 / alpha);
	} else {
		tmp = fma(cbrt(pow(t_0, 2.0)), cbrt(t_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)
	t_0 = Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0)))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -1.0)
		tmp = Float64(Float64(beta / alpha) + Float64(1.0 / alpha));
	else
		tmp = Float64(fma(cbrt((t_0 ^ 2.0)), cbrt(t_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_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(beta / alpha), $MachinePrecision] + N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 1/3], $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)) < -1

   1. Initial program 5.3%

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

   2. Simplified 5.3%

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

      [Start] 5.3	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 5.3	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Taylor expanded in alpha around -inf 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{-1 \cdot \frac{-1 \cdot \beta - \left(\beta + 2\right)}{\alpha}}{2} \]
      associate-*r/ [=>] 100.0	\[ \frac{\color{blue}{\frac{-1 \cdot \left(-1 \cdot \beta - \left(\beta + 2\right)\right)}{\alpha}}}{2} \]
      sub-neg [=>] 100.0	\[ \frac{\frac{-1 \cdot \color{blue}{\left(-1 \cdot \beta + \left(-\left(\beta + 2\right)\right)\right)}}{\alpha}}{2} \]
      mul-1-neg [<=] 100.0	\[ \frac{\frac{-1 \cdot \left(-1 \cdot \beta + \color{blue}{-1 \cdot \left(\beta + 2\right)}\right)}{\alpha}}{2} \]
      distribute-lft-in [=>] 100.0	\[ \frac{\frac{\color{blue}{-1 \cdot \left(-1 \cdot \beta\right) + -1 \cdot \left(-1 \cdot \left(\beta + 2\right)\right)}}{\alpha}}{2} \]
      neg-mul-1 [<=] 100.0	\[ \frac{\frac{\color{blue}{\left(--1 \cdot \beta\right)} + -1 \cdot \left(-1 \cdot \left(\beta + 2\right)\right)}{\alpha}}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{\frac{\left(-\color{blue}{\left(-\beta\right)}\right) + -1 \cdot \left(-1 \cdot \left(\beta + 2\right)\right)}{\alpha}}{2} \]
      remove-double-neg [=>] 100.0	\[ \frac{\frac{\color{blue}{\beta} + -1 \cdot \left(-1 \cdot \left(\beta + 2\right)\right)}{\alpha}}{2} \]
      neg-mul-1 [<=] 100.0	\[ \frac{\frac{\beta + \color{blue}{\left(--1 \cdot \left(\beta + 2\right)\right)}}{\alpha}}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{\frac{\beta + \left(-\color{blue}{\left(-\left(\beta + 2\right)\right)}\right)}{\alpha}}{2} \]
      remove-double-neg [=>] 100.0	\[ \frac{\frac{\beta + \color{blue}{\left(\beta + 2\right)}}{\alpha}}{2} \]
      +-commutative [=>] 100.0	\[ \frac{\frac{\beta + \color{blue}{\left(2 + \beta\right)}}{\alpha}}{2} \]

   5. Taylor expanded in beta around 0 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.1%

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

   2. Simplified 99.1%

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

      [Start] 99.1	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 99.1	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}, \sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}, 1\right)}}{2} \]
      Proof

      [Start] 99.1	\[ \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2} \]
      add-cube-cbrt [=>] 99.0	\[ \frac{\color{blue}{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}} + 1}{2} \]
      fma-def [=>] 99.0	\[ \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, 1\right)}}{2} \]
      cbrt-unprod [=>] 99.1	\[ \frac{\mathsf{fma}\left(\color{blue}{\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}}, \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, 1\right)}{2} \]
      pow2 [=>] 99.1	\[ \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{2}}}, \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, 1\right)}{2} \]
      associate-+l+ [=>] 99.1	\[ \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)}^{2}}, \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, 1\right)}{2} \]
      div-inv [=>] 99.1	\[ \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}, \sqrt[3]{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\beta + \alpha\right) + 2}}}, 1\right)}{2} \]
      div-inv [<=] 99.1	\[ \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}, \sqrt[3]{\color{blue}{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}}, 1\right)}{2} \]
      associate-+l+ [=>] 99.1	\[ \frac{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}, \sqrt[3]{\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}}, 1\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	14276

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

Alternative 2
Accuracy	99.4%
Cost	1476

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

Alternative 3
Accuracy	72.8%
Cost	1108

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -2 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -1 \cdot 10^{-152}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq -2.8 \cdot 10^{-215}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -4.2 \cdot 10^{-244}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 820:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 4
Accuracy	73.3%
Cost	1108

\[\begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -2 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -1.5 \cdot 10^{-152}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq -1.55 \cdot 10^{-216}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -4 \cdot 10^{-244}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 5
Accuracy	67.0%
Cost	852

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -2.45 \cdot 10^{-111}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -9.4 \cdot 10^{-153}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq -1.05 \cdot 10^{-216}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -2.1 \cdot 10^{-244}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2.8:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 6
Accuracy	93.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + \frac{1}{\alpha}\\ \end{array} \]

Alternative 7
Accuracy	68.8%
Cost	324

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

Alternative 8
Accuracy	49.3%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023151 
(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))