Octave 3.8, jcobi/1

Average Error: 16.1 → 0.2

Time: 29.1s

Precision: binary64

Cost: 43716

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta - \alpha}{t_0}\\ t_2 := {t_1}^{3}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\alpha}{2}} + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {t_2}^{3}}{\left({t_1}^{2} + \left(1 + \frac{\alpha - \beta}{t_0}\right)\right) \cdot \left(t_2 \cdot t_2 + \left(1 - t_2\right)\right)}}{2}\\ \end{array} \]

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))
        (t_2 (pow t_1 3.0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
     (/ (+ (/ beta (/ alpha 2.0)) (/ 2.0 alpha)) 2.0)
     (/
      (/
       (+ 1.0 (pow t_2 3.0))
       (*
        (+ (pow t_1 2.0) (+ 1.0 (/ (- alpha beta) t_0)))
        (+ (* t_2 t_2) (- 1.0 t_2))))
      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 t_2 = pow(t_1, 3.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = ((beta / (alpha / 2.0)) + (2.0 / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + pow(t_2, 3.0)) / ((pow(t_1, 2.0) + (1.0 + ((alpha - beta) / t_0))) * ((t_2 * t_2) + (1.0 - t_2)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = (beta - alpha) / t_0
    t_2 = t_1 ** 3.0d0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999d0)) then
        tmp = ((beta / (alpha / 2.0d0)) + (2.0d0 / alpha)) / 2.0d0
    else
        tmp = ((1.0d0 + (t_2 ** 3.0d0)) / (((t_1 ** 2.0d0) + (1.0d0 + ((alpha - beta) / t_0))) * ((t_2 * t_2) + (1.0d0 - t_2)))) / 2.0d0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = (beta - alpha) / t_0;
	double t_2 = Math.pow(t_1, 3.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = ((beta / (alpha / 2.0)) + (2.0 / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + Math.pow(t_2, 3.0)) / ((Math.pow(t_1, 2.0) + (1.0 + ((alpha - beta) / t_0))) * ((t_2 * t_2) + (1.0 - t_2)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

↓

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = (beta - alpha) / t_0
	t_2 = math.pow(t_1, 3.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999:
		tmp = ((beta / (alpha / 2.0)) + (2.0 / alpha)) / 2.0
	else:
		tmp = ((1.0 + math.pow(t_2, 3.0)) / ((math.pow(t_1, 2.0) + (1.0 + ((alpha - beta) / t_0))) * ((t_2 * t_2) + (1.0 - t_2)))) / 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)
	t_2 = t_1 ^ 3.0
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(Float64(beta / Float64(alpha / 2.0)) + Float64(2.0 / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + (t_2 ^ 3.0)) / Float64(Float64((t_1 ^ 2.0) + Float64(1.0 + Float64(Float64(alpha - beta) / t_0))) * Float64(Float64(t_2 * t_2) + Float64(1.0 - t_2)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

↓

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = (beta - alpha) / t_0;
	t_2 = t_1 ^ 3.0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999)
		tmp = ((beta / (alpha / 2.0)) + (2.0 / alpha)) / 2.0;
	else
		tmp = ((1.0 + (t_2 ^ 3.0)) / (((t_1 ^ 2.0) + (1.0 + ((alpha - beta) / t_0))) * ((t_2 * t_2) + (1.0 - t_2)))) / 2.0;
	end
	tmp_2 = 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]}, Block[{t$95$2 = N[Power[t$95$1, 3.0], $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999], N[(N[(N[(beta / N[(alpha / 2.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(1.0 + N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(1.0 - t$95$2), $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)) < -0.999990000000000046

   1. Initial program 59.6

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

   2. Simplified 59.6

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

      [Start] 59.6	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 59.6	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 59.6

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

   4. Simplified 59.6

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

      [Start] 59.6	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
      +-commutative [=>] 59.6	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\beta + \color{blue}{\left(2 + \alpha\right)}}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
      +-commutative [=>] 59.6	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
      +-commutative [<=] 59.6	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}}{2} \]
      +-commutative [=>] 59.6	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\beta + \color{blue}{\left(2 + \alpha\right)}}}}{2} \]
      +-commutative [=>] 59.6	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}}}{2} \]
      +-commutative [<=] 59.6	\[ \frac{\frac{1 - {\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}^{2}}{1 - \frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}}}{2} \]

   5. Taylor expanded in alpha around inf 0.6

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

   6. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\frac{\beta - -1 \cdot \left(\beta + 2\right)}{\alpha}}{2} \]
      distribute-lft-in [=>] 0.6	\[ \frac{\frac{\beta - \color{blue}{\left(-1 \cdot \beta + -1 \cdot 2\right)}}{\alpha}}{2} \]
      mul-1-neg [=>] 0.6	\[ \frac{\frac{\beta - \left(\color{blue}{\left(-\beta\right)} + -1 \cdot 2\right)}{\alpha}}{2} \]
      metadata-eval [=>] 0.6	\[ \frac{\frac{\beta - \left(\left(-\beta\right) + \color{blue}{-2}\right)}{\alpha}}{2} \]

   7. Applied egg-rr 0.6

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
      +-commutative [=>] 0.1	\[ \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Applied egg-rr 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\alpha}{2}} + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}^{3}}{\left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(1 + \frac{\alpha - \beta}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left({\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3} \cdot {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3} + \left(1 - {\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	21828

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\alpha}{2}} + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\frac{\frac{\beta - \alpha}{\frac{t_0}{\beta - \alpha}}}{t_0}}, \sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\alpha}{2}} + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1476

\[\begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\alpha}{2}} + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \]

Alternative 4
Error	4.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 125000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\frac{\alpha}{2}} + \frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	27.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 20500000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \end{array} \]

Alternative 6
Error	4.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 62000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} \cdot \left(\beta + 1\right)}{2}\\ \end{array} \]

Alternative 7
Error	30.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 330000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	40.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022354 
(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))