Octave 3.8, jcobi/4

Average Error: 54.0 → 34.9

Time: 23.7s

Precision: binary64

Cost: 41156

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

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

Math

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

↓

\[\begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_3 := \alpha + \left(\beta + i\right)\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_2}{t_0}}}{t_2 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{t_2}}{t_2 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left({\left(e^{\mathsf{fma}\left(i, t_3, \beta \cdot \alpha\right)}\right)}^{\left(\frac{i}{{t_2}^{2}} \cdot t_3\right)}\right)}{-1 + t_1 \cdot t_1}\\ \end{array} \]

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 (+ i (+ beta alpha)))
        (t_1 (+ (+ beta alpha) (* i 2.0)))
        (t_2 (fma i 2.0 (+ beta alpha)))
        (t_3 (+ alpha (+ beta i))))
   (if (<= beta 4.2e+222)
     (*
      (/ (/ i (/ t_2 t_0)) (+ t_2 1.0))
      (/ (/ (fma i t_0 (* beta alpha)) t_2) (+ t_2 -1.0)))
     (/
      (log (pow (exp (fma i t_3 (* beta alpha))) (* (/ i (pow t_2 2.0)) t_3)))
      (+ -1.0 (* t_1 t_1))))))

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 = i + (beta + alpha);
	double t_1 = (beta + alpha) + (i * 2.0);
	double t_2 = fma(i, 2.0, (beta + alpha));
	double t_3 = alpha + (beta + i);
	double tmp;
	if (beta <= 4.2e+222) {
		tmp = ((i / (t_2 / t_0)) / (t_2 + 1.0)) * ((fma(i, t_0, (beta * alpha)) / t_2) / (t_2 + -1.0));
	} else {
		tmp = log(pow(exp(fma(i, t_3, (beta * alpha))), ((i / pow(t_2, 2.0)) * t_3))) / (-1.0 + (t_1 * t_1));
	}
	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(i + Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_2 = fma(i, 2.0, Float64(beta + alpha))
	t_3 = Float64(alpha + Float64(beta + i))
	tmp = 0.0
	if (beta <= 4.2e+222)
		tmp = Float64(Float64(Float64(i / Float64(t_2 / t_0)) / Float64(t_2 + 1.0)) * Float64(Float64(fma(i, t_0, Float64(beta * alpha)) / t_2) / Float64(t_2 + -1.0)));
	else
		tmp = Float64(log((exp(fma(i, t_3, Float64(beta * alpha))) ^ Float64(Float64(i / (t_2 ^ 2.0)) * t_3))) / Float64(-1.0 + Float64(t_1 * t_1)));
	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[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(alpha + N[(beta + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.2e+222], N[(N[(N[(i / N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Power[N[Exp[N[(i * t$95$3 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(i / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(-1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.2000000000000002e222

   1. Initial program 51.8

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

   2. Applied egg-rr 33.3

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

   if 4.2000000000000002e222 < beta

   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. Applied egg-rr 57.9

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

   3. Applied egg-rr 42.0

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

4. Final simplification 34.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left({\left(e^{\mathsf{fma}\left(i, \alpha + \left(\beta + i\right), \beta \cdot \alpha\right)}\right)}^{\left(\frac{i}{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}} \cdot \left(\alpha + \left(\beta + i\right)\right)\right)}\right)}{-1 + \left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	37.7
Cost	34880

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := i + \left(\beta + \alpha\right)\\ \frac{\frac{i}{\frac{t_0}{t_1}}}{t_0 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)}{t_0}}{t_0 + -1} \end{array} \]

Alternative 2
Error	39.7
Cost	34560

\[\begin{array}{l} t_0 := \beta + \left(i + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \frac{t_0}{t_1 + 1} \cdot \frac{\left(i \cdot {\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}^{-2}\right) \cdot \mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{t_1 + -1} \end{array} \]

Alternative 3
Error	39.7
Cost	34560

\[\begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \frac{i \cdot \left(t_0 \cdot {t_1}^{-2}\right)}{\frac{t_1 + 1}{\frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{t_1 + -1}}} \end{array} \]

Alternative 4
Error	39.8
Cost	22208

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := i + \left(\beta + \alpha\right)\\ \frac{\frac{i}{\frac{t_1}{t_2}} \cdot \frac{\mathsf{fma}\left(i, t_2, \beta \cdot \alpha\right)}{t_1}}{-1 + t_0 \cdot t_0} \end{array} \]

Alternative 5
Error	50.3
Cost	21888

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := i + \left(\beta + \alpha\right)\\ \frac{i \cdot \left({\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{-2} \cdot \left(t_1 \cdot \mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)\right)\right)}{-1 + t_0 \cdot t_0} \end{array} \]

Alternative 6
Error	39.8
Cost	21888

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := i + \left(\beta + \alpha\right)\\ \frac{i \cdot \frac{t_1}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)}}}{-1 + t_0 \cdot t_0} \end{array} \]

Alternative 7
Error	50.3
Cost	16000

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := \beta + \left(i + \alpha\right)\\ \frac{\frac{\frac{i}{\frac{i \cdot -2 - \left(\beta + \alpha\right)}{t_1 \cdot \left(-\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right)\right)}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{-1 + t_0 \cdot t_0} \end{array} \]

Alternative 8
Error	54.1
Cost	3392

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) + i \cdot 2\\ t_1 := \alpha + \left(\beta + i\right)\\ t_2 := t_0 \cdot t_0\\ \frac{\frac{i \cdot \left(t_1 \cdot \left(\beta \cdot \alpha + i \cdot t_1\right)\right)}{t_2}}{-1 + t_2} \end{array} \]

Alternative 9
Error	54.0
Cost	3392

\[\begin{array}{l} t_0 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ t_1 := \left(\beta + \alpha\right) + i \cdot 2\\ t_2 := t_1 \cdot t_1\\ \frac{\frac{t_0 \cdot \left(\beta \cdot \alpha + t_0\right)}{t_2}}{-1 + t_2} \end{array} \]

Reproduce

herbie shell --seed 2022334 
(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)))