a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 8.8s

Precision: binary64

Cost: 7040

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]

↓

\[\mathsf{fma}\left(\frac{\left(1 - m\right) \cdot m}{v}, m, -m\right) \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

↓

(FPCore (m v) :precision binary64 (fma (/ (* (- 1.0 m) m) v) m (- m)))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

↓

double code(double m, double v) {
	return fma((((1.0 - m) * m) / v), m, -m);
}

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

↓

function code(m, v)
	return fma(Float64(Float64(Float64(1.0 - m) * m) / v), m, Float64(-m))
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

↓

code[m_, v_] := N[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] * m + (-m)), $MachinePrecision]

Derivation

1. Initial program 0.2

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right) \cdot m} \]
   Proof

3. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(1 - m\right) \cdot m}{v}, m, -m\right)} \]

Alternatives

Alternative 1
Error	0.2
Cost	768

\[\frac{m \cdot \left(1 - m\right)}{v} \cdot m + \left(-m\right) \]

Alternative 2
Error	25.7
Cost	716

\[\begin{array}{l} t_0 := \frac{m}{v} \cdot m\\ \mathbf{if}\;m \leq 5.881790461856417 \cdot 10^{-216}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.5192872812718196 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 6.116716548626432 \cdot 10^{-127}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 6.5105222143636235 \cdot 10^{-18}:\\ \;\;\;\;\frac{m}{v} \cdot m + \left(-m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 6.5105222143636235 \cdot 10^{-18}:\\ \;\;\;\;\frac{m}{v} \cdot m + \left(-m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 5
Error	0.2
Cost	704

\[\left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \cdot m \]

Alternative 6
Error	0.2
Cost	704

\[\frac{m \cdot \left(1 - m\right) - v}{v} \cdot m \]

Alternative 7
Error	2.3
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03358259997196748:\\ \;\;\;\;\frac{m}{v} \cdot m + \left(-m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-m \cdot m\right) \cdot m}{v}\\ \end{array} \]

Alternative 8
Error	2.3
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03358259997196748:\\ \;\;\;\;\frac{m}{v} \cdot m + \left(-m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \left(-\frac{m}{v}\right)\\ \end{array} \]

Alternative 9
Error	10.0
Cost	512

\[\frac{m}{v} \cdot m + \left(-m\right) \]

Alternative 10
Error	10.0
Cost	448

\[\frac{m - v}{v} \cdot m \]

Alternative 11
Error	36.3
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022320 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))