Octave 3.8, jcobi/1

Average Error: 16.1 → 0.1

Time: 7.9s

Precision: binary64

Cost: 8964

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta + 2}{\alpha}\\ t_1 := \beta + \left(\alpha + 2\right)\\ t_2 := \frac{\beta}{t_1}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{t_2 - \left(t_0 \cdot \left(t_0 + -1\right) - {t_0}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 + \left(1 - \frac{\alpha}{t_1}\right)}{2}\\ \end{array} \]

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 2.0) alpha))
        (t_1 (+ beta (+ alpha 2.0)))
        (t_2 (/ beta t_1)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9)
     (/ (- t_2 (- (* t_0 (+ t_0 -1.0)) (pow t_0 3.0))) 2.0)
     (/ (+ t_2 (- 1.0 (/ alpha t_1))) 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 + 2.0) / alpha;
	double t_1 = beta + (alpha + 2.0);
	double t_2 = beta / t_1;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9) {
		tmp = (t_2 - ((t_0 * (t_0 + -1.0)) - pow(t_0, 3.0))) / 2.0;
	} else {
		tmp = (t_2 + (1.0 - (alpha / t_1))) / 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 + 2.0d0) / alpha
    t_1 = beta + (alpha + 2.0d0)
    t_2 = beta / t_1
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9d0)) then
        tmp = (t_2 - ((t_0 * (t_0 + (-1.0d0))) - (t_0 ** 3.0d0))) / 2.0d0
    else
        tmp = (t_2 + (1.0d0 - (alpha / t_1))) / 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 + 2.0) / alpha;
	double t_1 = beta + (alpha + 2.0);
	double t_2 = beta / t_1;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9) {
		tmp = (t_2 - ((t_0 * (t_0 + -1.0)) - Math.pow(t_0, 3.0))) / 2.0;
	} else {
		tmp = (t_2 + (1.0 - (alpha / t_1))) / 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 + 2.0) / alpha
	t_1 = beta + (alpha + 2.0)
	t_2 = beta / t_1
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9:
		tmp = (t_2 - ((t_0 * (t_0 + -1.0)) - math.pow(t_0, 3.0))) / 2.0
	else:
		tmp = (t_2 + (1.0 - (alpha / t_1))) / 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 + 2.0) / alpha)
	t_1 = Float64(beta + Float64(alpha + 2.0))
	t_2 = Float64(beta / t_1)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9)
		tmp = Float64(Float64(t_2 - Float64(Float64(t_0 * Float64(t_0 + -1.0)) - (t_0 ^ 3.0))) / 2.0);
	else
		tmp = Float64(Float64(t_2 + Float64(1.0 - Float64(alpha / t_1))) / 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 + 2.0) / alpha;
	t_1 = beta + (alpha + 2.0);
	t_2 = beta / t_1;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9)
		tmp = (t_2 - ((t_0 * (t_0 + -1.0)) - (t_0 ^ 3.0))) / 2.0;
	else
		tmp = (t_2 + (1.0 - (alpha / t_1))) / 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[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(beta / t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9], N[(N[(t$95$2 - N[(N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$2 + N[(1.0 - N[(alpha / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.900000000000000022

   1. Initial program 58.8

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

   2. Applied egg-rr 56.8

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

   3. Taylor expanded in alpha around inf 6.4

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(-1 \cdot \frac{\beta + 2}{\alpha} + \left(\frac{{\left(\beta + 2\right)}^{2}}{{\alpha}^{2}} + -1 \cdot \frac{{\left(\beta + 2\right)}^{3}}{{\alpha}^{3}}\right)\right)}}{2} \]

   4. Simplified 0.4

      \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \color{blue}{\left(\frac{\beta + 2}{\alpha} \cdot \left(-1 + \frac{\beta + 2}{\alpha}\right) - {\left(\frac{\beta + 2}{\alpha}\right)}^{3}\right)}}{2} \]
      Proof
      (-.f64 (*.f64 (/.f64 (+.f64 beta 2) alpha) (+.f64 -1 (/.f64 (+.f64 beta 2) alpha))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 beta 2) alpha)))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta 2)) (*.f64 alpha alpha)))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 7 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta 2) 2)) (*.f64 alpha alpha))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2)))) (pow.f64 (/.f64 (+.f64 beta 2) alpha) 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (Rewrite<= cube-unmult_binary64 (*.f64 (/.f64 (+.f64 beta 2) alpha) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (+.f64 beta 2) alpha))))): 3 points increase in error, 6 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta 2) alpha) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta 2) (+.f64 beta 2)) (*.f64 alpha alpha))))): 9 points increase in error, 11 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (+.f64 beta 2) 2)) (*.f64 alpha alpha)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (*.f64 (/.f64 (+.f64 beta 2) alpha) (/.f64 (pow.f64 (+.f64 beta 2) 2) (Rewrite<= unpow2_binary64 (pow.f64 alpha 2))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 beta 2) (pow.f64 (+.f64 beta 2) 2)) (*.f64 alpha (pow.f64 alpha 2))))): 12 points increase in error, 17 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (*.f64 (+.f64 beta 2) (Rewrite=> unpow2_binary64 (*.f64 (+.f64 beta 2) (+.f64 beta 2)))) (*.f64 alpha (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (Rewrite<= cube-mult_binary64 (pow.f64 (+.f64 beta 2) 3)) (*.f64 alpha (pow.f64 alpha 2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (pow.f64 (+.f64 beta 2) 3) (*.f64 alpha (Rewrite=> unpow2_binary64 (*.f64 alpha alpha))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (/.f64 (pow.f64 (+.f64 beta 2) 3) (Rewrite<= cube-mult_binary64 (pow.f64 alpha 3)))): 6 points increase in error, 4 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (neg.f64 (/.f64 (pow.f64 (+.f64 beta 2) 3) (pow.f64 alpha 3))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 3) (pow.f64 alpha 3))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 beta 2) alpha)) (+.f64 (/.f64 (pow.f64 (+.f64 beta 2) 2) (pow.f64 alpha 2)) (*.f64 -1 (/.f64 (pow.f64 (+.f64 beta 2) 3) (pow.f64 alpha 3)))))): 1 points increase in error, 0 points decrease in error

   if -0.900000000000000022 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\beta + 2}{\alpha} \cdot \left(\frac{\beta + 2}{\alpha} + -1\right) - {\left(\frac{\beta + 2}{\alpha}\right)}^{3}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	7876

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

Alternative 2
Error	0.4
Cost	1860

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

Alternative 3
Error	0.4
Cost	1476

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

Alternative 4
Error	4.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7735799440787.236:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\beta}{\alpha} + \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 5
Error	8.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7735799440787.236:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	4.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7735799440787.236:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	17.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.0020510767633913156:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	38.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.341043821216218 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha}\\ \end{array} \]

Alternative 9
Error	18.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.0020510767633913156:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	39.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022310 
(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))