b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 7.4s

Precision: binary64

Cost: 7104

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)} \]
   Proof
   (*.f64 (-.f64 1 m) (fma.f64 (-.f64 1 m) (/.f64 m v) -1)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 1 m) (fma.f64 (-.f64 1 m) (/.f64 m v) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 1 m) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (-.f64 1 m) (/.f64 m v)) 1))): 1 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 1 m) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 m v) (-.f64 1 m))) 1)): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 1 m) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (-.f64 1 m)) v)) 1)): 6 points increase in error, 4 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) (-.f64 1 m))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 4.414584399640488 \cdot 10^{-12}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	832

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

Alternative 3
Error	0.1
Cost	832

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

Alternative 4
Error	2.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.2313802414290612:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 5
Error	1.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.956951031242246:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m + -2\right)}{\frac{v}{m}}\\ \end{array} \]

Alternative 6
Error	1.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.956951031242246:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \]

Alternative 7
Error	2.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 1.956951031242246:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 8
Error	24.5
Cost	324

\[\begin{array}{l} \mathbf{if}\;v \leq 1.5043712775898053 \cdot 10^{-130}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9
Error	24.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;v \leq 1.5043712775898053 \cdot 10^{-130}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 10
Error	9.8
Cost	320

\[\frac{m}{v} + -1 \]

Alternative 11
Error	61.7
Cost	64

\[m \]

Alternative 12
Error	37.4
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022317 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))