Octave 3.8, jcobi/4

Average Error: 54.2 → 1.4

Time: 31.8s

Precision: binary64

Cost: 15556

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{i}{1 + t_1}}{\frac{t_1}{i + \left(\alpha + \beta\right)}} \cdot \left(\frac{i}{t_0 + -1} \cdot \frac{i + \beta}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\beta}\\ \end{array} \]

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 (+ beta (* i 2.0))) (t_1 (fma i 2.0 (+ alpha beta))))
   (if (<= alpha 2.4e+184)
     (*
      (/ (/ i (+ 1.0 t_1)) (/ t_1 (+ i (+ alpha beta))))
      (* (/ i (+ t_0 -1.0)) (/ (+ i beta) t_0)))
     (/ (* i (/ (+ alpha i) beta)) 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 = beta + (i * 2.0);
	double t_1 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (alpha <= 2.4e+184) {
		tmp = ((i / (1.0 + t_1)) / (t_1 / (i + (alpha + beta)))) * ((i / (t_0 + -1.0)) * ((i + beta) / t_0));
	} else {
		tmp = (i * ((alpha + i) / beta)) / beta;
	}
	return tmp;
}

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(beta + Float64(i * 2.0))
	t_1 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (alpha <= 2.4e+184)
		tmp = Float64(Float64(Float64(i / Float64(1.0 + t_1)) / Float64(t_1 / Float64(i + Float64(alpha + beta)))) * Float64(Float64(i / Float64(t_0 + -1.0)) * Float64(Float64(i + beta) / t_0)));
	else
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) / beta)) / beta);
	end
	return tmp
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[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.4e+184], N[(N[(N[(i / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.39999999999999997e184

   1. Initial program 54.0

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

   2. Applied egg-rr 37.9

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

   3. Taylor expanded in alpha around 0 40.0

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

   4. Simplified 1.2

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \color{blue}{\left(\frac{i}{\left(\beta + i \cdot 2\right) + -1} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right)} \]
      Proof
      (*.f64 (/.f64 i (+.f64 (+.f64 beta (*.f64 i 2)) -1)) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (+.f64 (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i))) -1)) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (+.f64 (+.f64 beta (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 beta (*.f64 2 i)) 1))) (/.f64 (+.f64 beta i) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 i (-.f64 (+.f64 beta (*.f64 2 i)) 1)) (/.f64 (+.f64 beta i) (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 beta i)) (*.f64 (-.f64 (+.f64 beta (*.f64 2 i)) 1) (+.f64 beta (*.f64 2 i))))): 176 points increase in error, 17 points decrease in error

   5. Applied egg-rr 12.1

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}\right)} - 1\right)} \cdot \left(\frac{i}{\left(\beta + i \cdot 2\right) + -1} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \]

   6. Simplified 1.2

      \[\leadsto \color{blue}{\frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}}} \cdot \left(\frac{i}{\left(\beta + i \cdot 2\right) + -1} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \]
      Proof
      (/.f64 (/.f64 i (+.f64 1 (fma.f64 i 2 (+.f64 beta alpha)))) (/.f64 (fma.f64 i 2 (+.f64 beta alpha)) (+.f64 (+.f64 beta alpha) i))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 i (+.f64 1 (fma.f64 i 2 (Rewrite=> +-commutative_binary64 (+.f64 alpha beta))))) (/.f64 (fma.f64 i 2 (+.f64 beta alpha)) (+.f64 (+.f64 beta alpha) i))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1))) (/.f64 (fma.f64 i 2 (+.f64 beta alpha)) (+.f64 (+.f64 beta alpha) i))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 i (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1)) (/.f64 (fma.f64 i 2 (Rewrite=> +-commutative_binary64 (+.f64 alpha beta))) (+.f64 (+.f64 beta alpha) i))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 i (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1)) (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (Rewrite<= +-commutative_binary64 (+.f64 i (+.f64 beta alpha))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 i (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1)) (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 i (Rewrite=> +-commutative_binary64 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 (/.f64 i (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1)) (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 i (+.f64 alpha beta))))))): 1 points increase in error, 0 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 (/.f64 i (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1)) (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 i (+.f64 alpha beta)))))) 1)): 42 points increase in error, 37 points decrease in error

   if 2.39999999999999997e184 < alpha

   1. Initial program 64.0

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

   2. Taylor expanded in beta around inf 60.0

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

   3. Simplified 57.9

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{i + \alpha}}} \]
      Proof
      (/.f64 i (/.f64 (*.f64 beta beta) (+.f64 i alpha))): 0 points increase in error, 0 points decrease in error
      (/.f64 i (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 beta 2)) (+.f64 i alpha))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 i (+.f64 i alpha)) (pow.f64 beta 2))): 83 points increase in error, 7 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 i alpha) i)) (pow.f64 beta 2)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 38.4

      \[\leadsto \frac{i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\beta}{\frac{i + \alpha}{\beta}}\right)\right)}} \]

   5. Applied egg-rr 14.5

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

   6. Applied egg-rr 14.5

      \[\leadsto \color{blue}{\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}} \cdot \left(\frac{i}{\left(\beta + i \cdot 2\right) + -1} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	9028

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+184}:\\ \;\;\;\;\left(\frac{i}{t_0 + -1} \cdot \frac{i + \beta}{t_0}\right) \cdot \frac{\frac{i}{\frac{t_0}{i + \beta}}}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\beta}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	2628

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+184}:\\ \;\;\;\;\left(\frac{i}{t_0 + -1} \cdot \frac{i + \beta}{t_0}\right) \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{i \cdot 2 + \left(1 + \beta\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta}}{\beta}\\ \end{array} \]

Alternative 3
Error	9.3
Cost	2500

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+199}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \beta}{i \cdot 2 + \left(1 + \beta\right)}\right) \cdot \left(\frac{i}{t_0 + -1} \cdot \left(0.5 + 0.25 \cdot \frac{\beta}{i}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternative 4
Error	9.3
Cost	2116

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 1.75 \cdot 10^{+199}:\\ \;\;\;\;\left(\frac{i}{t_0 + -1} \cdot \frac{i + \beta}{t_0}\right) \cdot \left(0.5 \cdot \frac{i}{1 + \left(\alpha + i \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternative 5
Error	9.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	9.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternative 7
Error	15.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+234}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 8
Error	10.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+199}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Error	18.1
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022321 
(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)))