Octave 3.8, jcobi/4

Average Error: 53.8 → 0.3

Time: 25.9s

Precision: binary64

Cost: 3264

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	return ((((i + alpha) / t_0) * (i + beta)) / (alpha + (beta + ((i * 2.0) + 1.0)))) * (i / ((t_0 + -1.0) * (t_0 / (i + (alpha + beta)))))

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 53.8

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

2. Applied egg-rr 53.8

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

3. Simplified 53.8

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

   [Start] 53.8	\[ \frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right) \cdot \left(i \cdot \left(\alpha + \beta\right)\right) + \mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right) \cdot \left(i \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   +-commutative [<=] 53.8	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right) \cdot \left(i \cdot i\right) + \mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right) \cdot \left(i \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   distribute-lft-out [=>] 53.8	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right) \cdot \left(i \cdot i + i \cdot \left(\alpha + \beta\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Applied egg-rr 0.3

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

5. Simplified 0.3

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

   [Start] 0.3	\[ \frac{\frac{i + \alpha}{\frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \beta}}}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{\frac{i}{\frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}}}{\left(\beta + \left(\alpha + i \cdot 2\right)\right) + -1} \]
   +-commutative [=>] 0.3	\[ \frac{\frac{i + \alpha}{\frac{\beta + \left(\alpha + i \cdot 2\right)}{\color{blue}{\beta + i}}}}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{\frac{i}{\frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}}}{\left(\beta + \left(\alpha + i \cdot 2\right)\right) + -1} \]
   associate-/r/ [=>] 0.3	\[ \frac{\color{blue}{\frac{i + \alpha}{\beta + \left(\alpha + i \cdot 2\right)} \cdot \left(\beta + i\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{\frac{i}{\frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}}}{\left(\beta + \left(\alpha + i \cdot 2\right)\right) + -1} \]
   associate-+r+ [=>] 0.3	\[ \frac{\frac{i + \alpha}{\color{blue}{\left(\beta + \alpha\right) + i \cdot 2}} \cdot \left(\beta + i\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{\frac{i}{\frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}}}{\left(\beta + \left(\alpha + i \cdot 2\right)\right) + -1} \]
   associate-+l+ [=>] 0.3	\[ \frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta + i\right)}{\alpha + \color{blue}{\left(\beta + \left(i \cdot 2 + 1\right)\right)}} \cdot \frac{\frac{i}{\frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}}}{\left(\beta + \left(\alpha + i \cdot 2\right)\right) + -1} \]
   associate-/l/ [=>] 0.3	\[ \frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta + i\right)}{\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)} \cdot \color{blue}{\frac{i}{\left(\left(\beta + \left(\alpha + i \cdot 2\right)\right) + -1\right) \cdot \frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}}} \]
   +-commutative [=>] 0.3	\[ \frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta + i\right)}{\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)} \cdot \frac{i}{\color{blue}{\left(-1 + \left(\beta + \left(\alpha + i \cdot 2\right)\right)\right)} \cdot \frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}} \]
   associate-+r+ [=>] 0.3	\[ \frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta + i\right)}{\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)} \cdot \frac{i}{\left(-1 + \color{blue}{\left(\left(\beta + \alpha\right) + i \cdot 2\right)}\right) \cdot \frac{\beta + \left(\alpha + i \cdot 2\right)}{i + \left(\alpha + \beta\right)}} \]
   associate-+r+ [=>] 0.3	\[ \frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta + i\right)}{\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)} \cdot \frac{i}{\left(-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)\right) \cdot \frac{\color{blue}{\left(\beta + \alpha\right) + i \cdot 2}}{i + \left(\alpha + \beta\right)}} \]
   +-commutative [<=] 0.3	\[ \frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \left(\beta + i\right)}{\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)} \cdot \frac{i}{\left(-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)\right) \cdot \frac{\left(\beta + \alpha\right) + i \cdot 2}{i + \color{blue}{\left(\beta + \alpha\right)}}} \]

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	2.0
Cost	3008

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

Alternative 2
Error	9.2
Cost	2628

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 3.15 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \beta}{t_1}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\left(t_0 + -1\right) \cdot \frac{t_0}{i + \left(\alpha + \beta\right)}} \cdot \frac{i + \alpha}{\alpha + \left(\beta + \left(i \cdot 2 + 1\right)\right)}\\ \end{array} \]

Alternative 3
Error	9.9
Cost	1348

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+137}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \beta}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+135}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	16.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.8 \cdot 10^{+254}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\alpha}{\beta \cdot \beta}\\ \end{array} \]

Alternative 6
Error	16.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+254}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 7
Error	16.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+203}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 8
Error	11.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Error	18.7
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023018 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))