Octave 3.8, jcobi/3

Average Error: 3.7 → 0.1

Time: 14.0s

Precision: binary64

Cost: 1600

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

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \frac{\frac{1 + \alpha}{t_0}}{\frac{t_0}{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)}}} \end{array} \]

FPCore

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

↓

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (/ (/ (+ 1.0 alpha) t_0) (/ t_0 (/ (+ 1.0 beta) (+ beta (+ alpha 3.0)))))))

C

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

↓

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return ((1.0 + alpha) / t_0) / (t_0 / ((1.0 + beta) / (beta + (alpha + 3.0))));
}

Fortran

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

↓

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = beta + (alpha + 2.0d0)
    code = ((1.0d0 + alpha) / t_0) / (t_0 / ((1.0d0 + beta) / (beta + (alpha + 3.0d0))))
end function

Java

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

↓

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	return ((1.0 + alpha) / t_0) / (t_0 / ((1.0 + beta) / (beta + (alpha + 3.0))));
}

Python

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

↓

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	return ((1.0 + alpha) / t_0) / (t_0 / ((1.0 + beta) / (beta + (alpha + 3.0))))

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * 1.0)) + 1.0))
end

↓

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	return Float64(Float64(Float64(1.0 + alpha) / t_0) / Float64(t_0 / Float64(Float64(1.0 + beta) / Float64(beta + Float64(alpha + 3.0)))))
end

MATLAB

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

↓

function tmp = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = ((1.0 + alpha) / t_0) / (t_0 / ((1.0 + beta) / (beta + (alpha + 3.0))));
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 + alpha), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 / N[(N[(1.0 + beta), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 3.7

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

2. Simplified 10.0

   \[\leadsto \color{blue}{\frac{\left(\alpha + 1\right) \cdot \left(\beta + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   Proof
   (/.f64 (*.f64 (+.f64 alpha 1) (+.f64 beta 1)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 alpha)) (+.f64 beta 1)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta alpha) 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 beta alpha) 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 alpha beta)) 2) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (+.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) 2) 1))))): 0 points increase in error, 1 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (+.f64 (+.f64 (+.f64 alpha beta) (Rewrite<= metadata-eval (*.f64 2 1))) 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (+.f64 1 alpha) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (/.f64 (+.f64 beta 1) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))))): 14 points increase in error, 44 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 beta 1)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))): 9 points increase in error, 16 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 beta 1) (/.f64 (+.f64 1 alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 1))))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 beta 1) (+.f64 1 alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 18 points increase in error, 2 points decrease in error
   (/.f64 (/.f64 (*.f64 (+.f64 beta 1) (Rewrite=> +-commutative_binary64 (+.f64 alpha 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (*.f64 (+.f64 beta 1) 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 2 points increase in error, 1 points decrease in error
   (/.f64 (/.f64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (Rewrite=> *-rgt-identity_binary64 (+.f64 beta 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 beta alpha) alpha)) (+.f64 beta 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 beta alpha) alpha) beta) 1)) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 beta alpha) (+.f64 alpha beta))) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha))) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 1) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 alpha beta) (*.f64 2 1))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 1)) 1))): 13 points increase in error, 27 points decrease in error

3. Applied egg-rr 2.4

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

   \[\leadsto \frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\frac{\beta + \left(\alpha + 2\right)}{\frac{1 + \beta}{\beta + \left(\alpha + 3\right)}}} \]

Alternatives

Alternative 1
Error	1.4
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 0.00048459706752260206:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(4 + \alpha \cdot 2\right)}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 11588.021311753486:\\ \;\;\;\;\frac{\frac{\frac{1 + \beta}{\beta + 2}}{2 + \left(\alpha + \beta\right)}}{\beta + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta + \left(4 + \alpha \cdot 2\right)}\\ \end{array} \]

Alternative 3
Error	1.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5821380048914895:\\ \;\;\;\;\frac{1 + \alpha}{\alpha + \left(\beta + 2\right)} \cdot \left(0.16666666666666666 + \alpha \cdot -0.1388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5821380048914895:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{1 + \left(2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \]

Alternative 5
Error	1.8
Cost	964

\[\begin{array}{l} t_0 := 1 + \left(2 + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\beta \leq 3.5821380048914895:\\ \;\;\;\;\frac{\frac{0.5}{\alpha + 2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 6
Error	2.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5821380048914895:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 2\right)}}{\beta}\\ \end{array} \]

Alternative 7
Error	4.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5821380048914895:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{elif}\;\beta \leq 7.311727508928696 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 8
Error	2.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5821380048914895:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	16.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.078625124543527 \cdot 10^{+19}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 10
Error	14.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.078625124543527 \cdot 10^{+19}:\\ \;\;\;\;\frac{0.16666666666666666}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	34.2
Cost	320

\[\frac{0.16666666666666666}{\beta + 2} \]

Alternative 12
Error	60.2
Cost	192

\[\frac{0.5}{\beta} \]

Reproduce

herbie shell --seed 2022315 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))