Octave 3.8, jcobi/3

Percentage Accurate: 99.8% → 99.8%

Time: 34.7s

Precision: binary64

Cost: 1856

• Unsound rule application detected in e-graph. Results may not be sound.

• could not determine a ground truth

  1. alpha = 1.6491314408836382e+119
  2. beta = 8.619254780095334e+207

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

Math

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

↓

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

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 (+ (+ alpha beta) 2.0)))
   (/ (/ (/ (+ (+ (+ alpha beta) (* alpha beta)) 1.0) t_0) t_0) (+ 1.0 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 = (alpha + beta) + 2.0;
	return (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + 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 = (alpha + beta) + 2.0d0
    code = (((((alpha + beta) + (alpha * beta)) + 1.0d0) / t_0) / t_0) / (1.0d0 + 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 = (alpha + beta) + 2.0;
	return (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + 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 = (alpha + beta) + 2.0
	return (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + 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(Float64(alpha + beta) + 2.0)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(alpha * beta)) + 1.0) / t_0) / t_0) / Float64(1.0 + 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 = (alpha + beta) + 2.0;
	tmp = (((((alpha + beta) + (alpha * beta)) + 1.0) / t_0) / t_0) / (1.0 + 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[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1856

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

Alternative 2
Accuracy	95.7%
Cost	1600

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

Alternative 3
Accuracy	95.7%
Cost	1600

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

Alternative 4
Accuracy	86.5%
Cost	1472

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

Alternative 5
Accuracy	93.6%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;\left(\alpha + 1\right) \cdot \frac{\frac{1}{\alpha + 2}}{\left(\alpha + 2\right) \cdot \left(\alpha + 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]

Alternative 6
Accuracy	73.3%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{1 + \left(\left(\alpha + \beta\right) + 2\right)}\\ \end{array} \]

Alternative 7
Accuracy	72.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \left(\beta + 3\right)}\\ \end{array} \]

Alternative 8
Accuracy	73.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + 1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 9
Accuracy	72.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\beta}}{\beta + 3}\\ \end{array} \]

Alternative 10
Accuracy	73.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + 1}{\beta}}{\beta}\\ \end{array} \]

Alternative 11
Accuracy	72.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \]

Alternative 12
Accuracy	50.5%
Cost	64

\[0.08333333333333333 \]

Reproduce

herbie shell --seed 2023277 
(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)))