Octave 3.8, jcobi/3

Average Error: 3.9 → 0.1

Time: 17.2s

Precision: binary64

Cost: 1600

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

Math

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

↓

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

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))))
   (/ (/ (/ (+ beta 1.0) (/ t_0 (+ 1.0 alpha))) (+ beta (+ alpha 3.0))) t_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 (((beta + 1.0) / (t_0 / (1.0 + alpha))) / (beta + (alpha + 3.0))) / t_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 = (((beta + 1.0d0) / (t_0 / (1.0d0 + alpha))) / (beta + (alpha + 3.0d0))) / t_0
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 (((beta + 1.0) / (t_0 / (1.0 + alpha))) / (beta + (alpha + 3.0))) / t_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 (((beta + 1.0) / (t_0 / (1.0 + alpha))) / (beta + (alpha + 3.0))) / t_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(Float64(beta + 1.0) / Float64(t_0 / Float64(1.0 + alpha))) / Float64(beta + Float64(alpha + 3.0))) / t_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 = (((beta + 1.0) / (t_0 / (1.0 + alpha))) / (beta + (alpha + 3.0))) / t_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[(N[(beta + 1.0), $MachinePrecision] / N[(t$95$0 / N[(1.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(alpha + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 3.9

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

2. Simplified 9.8

   \[\leadsto \color{blue}{\frac{\left(\beta + 1\right) \cdot \left(\alpha + 1\right)}{\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\beta + \left(\alpha + 2\right)\right) \cdot \left(\left(\alpha + \beta\right) + 3\right)\right)}} \]
   Proof
   (/.f64 (*.f64 (+.f64 beta 1) (+.f64 alpha 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 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (*.f64 (+.f64 beta 1) 1))) (*.f64 (+.f64 beta (+.f64 alpha 2)) (*.f64 (+.f64 beta (+.f64 alpha 2)) (+.f64 (+.f64 alpha beta) 3)))): 0 points increase in error, 1 points decrease in error
   (/.f64 (+.f64 (*.f64 (+.f64 beta 1) alpha) (Rewrite=> *-rgt-identity_binary64 (+.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<= distribute-lft1-in_binary64 (+.f64 (*.f64 beta alpha) 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 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 beta alpha) alpha) 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<= associate-+r+_binary64 (+.f64 (*.f64 beta alpha) (+.f64 alpha 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 (+.f64 alpha beta) (*.f64 beta alpha))) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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))))): 3 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 beta alpha)) 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 (+.f64 alpha beta) (*.f64 beta alpha)) 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=> associate-/r*_binary64 (/.f64 (/.f64 (+.f64 (+.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))))): 7 points increase in error, 31 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, 13 points decrease in error

3. Applied egg-rr 2.4

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	2.6
Cost	1732

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 1.22 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 + \alpha}{t_1} \cdot \frac{\beta + 1}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	1732

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 3\right)\\ t_1 := \alpha + \left(\beta + 2\right)\\ \mathbf{if}\;\beta \leq 10^{+154}:\\ \;\;\;\;\frac{\beta + 1}{t_1} \cdot \frac{1 + \alpha}{t_0 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta + 1}{\beta \cdot \frac{1}{1 + \alpha}}}{t_0}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	1600

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

Alternative 4
Error	4.7
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.95:\\ \;\;\;\;\frac{\frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 5
Error	4.7
Cost	1220

\[\begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 3:\\ \;\;\;\;\frac{\frac{1 + \alpha}{4 + \alpha \cdot \left(\alpha + 4\right)}}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{t_0}}{1 + t_0}\\ \end{array} \]

Alternative 6
Error	18.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.95:\\ \;\;\;\;\frac{0.25 + -0.0625 \cdot \left(\alpha \cdot \alpha\right)}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta + \left(\alpha + 3\right)}}{\beta + \left(\alpha + 2\right)}\\ \end{array} \]

Alternative 7
Error	18.9
Cost	964

\[\begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.95:\\ \;\;\;\;\frac{0.25 + -0.0625 \cdot \left(\alpha \cdot \alpha\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 8
Error	18.9
Cost	964

\[\begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\frac{0.25 + -0.0625 \cdot \left(\alpha \cdot \alpha\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta} + \frac{\alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 9
Error	18.1
Cost	836

\[\begin{array}{l} t_0 := 3 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.8:\\ \;\;\;\;\frac{0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \alpha}{\beta}}{t_0}\\ \end{array} \]

Alternative 10
Error	18.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Error	18.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta} \cdot \frac{1 + \alpha}{\beta}\\ \end{array} \]

Alternative 12
Error	19.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 13
Error	19.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.1:\\ \;\;\;\;\frac{0.25}{3 + \left(\beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 1.1 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 14
Error	34.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\beta}\\ \end{array} \]

Alternative 15
Error	20.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 16
Error	20.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.75:\\ \;\;\;\;0.08333333333333333 + \beta \cdot -0.027777777777777776\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta}\\ \end{array} \]

Alternative 17
Error	34.8
Cost	320

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

Alternative 18
Error	34.6
Cost	320

\[\frac{0.25}{\beta + 3} \]

Alternative 19
Error	35.5
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2022328 
(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)))