Average Error: 0.1 → 0.1
Time: 8.8s
Precision: binary64
Cost: 7104
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[\left(1 - m\right) \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right) \]
(FPCore (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)))
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);
}

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

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]

\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)

Error

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)} \]
    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))): 0 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)): 7 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 simplification0.1

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

Alternatives

Alternative 1
Error18.8
Cost976
\[\begin{array}{l} \mathbf{if}\;m \leq 1.3415489904602779 \cdot 10^{-204}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.7213138345374907 \cdot 10^{-154}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 2.5856999665715205 \cdot 10^{-142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
Alternative 2
Error18.7
Cost976
\[\begin{array}{l} \mathbf{if}\;m \leq 1.3415489904602779 \cdot 10^{-204}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.7213138345374907 \cdot 10^{-154}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 2.5856999665715205 \cdot 10^{-142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
Alternative 3
Error18.7
Cost976
\[\begin{array}{l} \mathbf{if}\;m \leq 1.3415489904602779 \cdot 10^{-204}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.7213138345374907 \cdot 10^{-154}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 2.5856999665715205 \cdot 10^{-142}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
Alternative 4
Error0.2
Cost836
\[\begin{array}{l} \mathbf{if}\;m \leq 7.801976606406805 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m + -1\right) \cdot \left(m \cdot \left(m + -1\right)\right)}{v}\\ \end{array} \]
Alternative 5
Error0.2
Cost832
\[\left(1 - m\right) \cdot \left(-1 + \frac{1 - m}{\frac{v}{m}}\right) \]
Alternative 6
Error0.1
Cost832
\[\left(1 - m\right) \cdot \left(-1 - \frac{m \cdot \left(m + -1\right)}{v}\right) \]
Alternative 7
Error1.8
Cost708
\[\begin{array}{l} \mathbf{if}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(m + -2\right)\right)\\ \end{array} \]
Alternative 8
Error1.8
Cost708
\[\begin{array}{l} \mathbf{if}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(m + -2\right)\right)}{v}\\ \end{array} \]
Alternative 9
Error1.8
Cost708
\[\begin{array}{l} \mathbf{if}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \]
Alternative 10
Error1.8
Cost708
\[\begin{array}{l} \mathbf{if}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -2\right)\\ \end{array} \]
Alternative 11
Error24.1
Cost588
\[\begin{array}{l} \mathbf{if}\;v \leq 2.066442605194484 \cdot 10^{-185}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;v \leq 5.1508927934001276 \cdot 10^{-169}:\\ \;\;\;\;-1\\ \mathbf{elif}\;v \leq 2.233684286384136 \cdot 10^{-141}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
Alternative 12
Error23.8
Cost588
\[\begin{array}{l} \mathbf{if}\;v \leq 2.066442605194484 \cdot 10^{-185}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;v \leq 5.1508927934001276 \cdot 10^{-169}:\\ \;\;\;\;m + -1\\ \mathbf{elif}\;v \leq 2.233684286384136 \cdot 10^{-141}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]
Alternative 13
Error2.4
Cost580
\[\begin{array}{l} \mathbf{if}\;m \leq 0.0025375023412834603:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]
Alternative 14
Error37.0
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022313 
(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)))