Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 10.3s

Precision: binary64

Cost: 13504

Math

\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

↓

\[\left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}\right) \]

FPCore

(FPCore (a rand)
 :precision binary64
 (*
  (- a (/ 1.0 3.0))
  (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))

↓

(FPCore (a rand)
 :precision binary64
 (* (- a 0.3333333333333333) (+ 1.0 (/ rand (sqrt (fma a 9.0 -3.0))))))

C

double code(double a, double rand) {
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
}

↓

double code(double a, double rand) {
	return (a - 0.3333333333333333) * (1.0 + (rand / sqrt(fma(a, 9.0, -3.0))));
}

Julia

function code(a, rand)
	return Float64(Float64(a - Float64(1.0 / 3.0)) * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * Float64(a - Float64(1.0 / 3.0))))) * rand)))
end

↓

function code(a, rand)
	return Float64(Float64(a - 0.3333333333333333) * Float64(1.0 + Float64(rand / sqrt(fma(a, 9.0, -3.0)))))
end

Wolfram

code[a_, rand_] := N[(N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, rand_] := N[(N[(a - 0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(rand / N[Sqrt[N[(a * 9.0 + -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}\right)} \]
   Proof
   (*.f64 (-.f64 a 1/3) (+.f64 1 (/.f64 rand (sqrt.f64 (fma.f64 a 9 -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (Rewrite<= metadata-eval (/.f64 1 3))) (+.f64 1 (/.f64 rand (sqrt.f64 (fma.f64 a 9 -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 rand)) (sqrt.f64 (fma.f64 a 9 -3))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (fma.f64 a 9 (Rewrite<= metadata-eval (*.f64 -1/3 9))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (fma.f64 a 9 (*.f64 (Rewrite<= metadata-eval (neg.f64 1/3)) 9)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (fma.f64 a 9 (*.f64 (neg.f64 (Rewrite<= metadata-eval (/.f64 1 3))) 9)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 a 9) (*.f64 (neg.f64 (/.f64 1 3)) 9))))))): 1 points increase in error, 1 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 9 (+.f64 a (neg.f64 (/.f64 1 3))))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (/.f64 (*.f64 1 rand) (sqrt.f64 (*.f64 9 (Rewrite<= sub-neg_binary64 (-.f64 a (/.f64 1 3)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)))): 11 points increase in error, 7 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 a (/.f64 1 3)) (Rewrite<= sub-neg_binary64 (-.f64 1 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1 (-.f64 a (/.f64 1 3))) (*.f64 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)) (-.f64 a (/.f64 1 3))))): 0 points increase in error, 1 points decrease in error
   (-.f64 (Rewrite=> *-lft-identity_binary64 (-.f64 a (/.f64 1 3))) (*.f64 (neg.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)) (-.f64 a (/.f64 1 3)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub_binary64 (+.f64 (-.f64 a (/.f64 1 3)) (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand) (-.f64 a (/.f64 1 3))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (-.f64 a (/.f64 1 3)))) (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand) (-.f64 a (/.f64 1 3)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-in_binary64 (*.f64 (-.f64 a (/.f64 1 3)) (+.f64 1 (*.f64 (/.f64 1 (sqrt.f64 (*.f64 9 (-.f64 a (/.f64 1 3))))) rand)))): 1 points increase in error, 0 points decrease in error

3. Final simplification 0.1

   \[\leadsto \left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{\mathsf{fma}\left(a, 9, -3\right)}}\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	7232

\[\left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{\frac{rand}{3}}{\sqrt{a - 0.3333333333333333}}\right) \]

Alternative 2
Error	5.8
Cost	7112

\[\begin{array}{l} t_0 := rand \cdot \sqrt{\left(a - 0.3333333333333333\right) \cdot 0.1111111111111111}\\ \mathbf{if}\;rand \leq -4.897079703193594 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 3.8482795150864534 \cdot 10^{+31}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	5.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;rand \leq -4.897079703193594 \cdot 10^{+74}:\\ \;\;\;\;rand \cdot \sqrt{\left(a - 0.3333333333333333\right) \cdot 0.1111111111111111}\\ \mathbf{elif}\;rand \leq 3.8482795150864534 \cdot 10^{+31}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - 0.3333333333333333} \cdot \left(0.3333333333333333 \cdot rand\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7104

\[\left(a - 0.3333333333333333\right) \cdot \left(1 + \frac{rand}{\sqrt{a \cdot 9}}\right) \]

Alternative 5
Error	0.2
Cost	7104

\[-0.3333333333333333 + \left(a + 0.3333333333333333 \cdot \left(rand \cdot \sqrt{a - 0.3333333333333333}\right)\right) \]

Alternative 6
Error	6.4
Cost	6984

\[\begin{array}{l} t_0 := rand \cdot \sqrt{a \cdot 0.1111111111111111}\\ \mathbf{if}\;rand \leq -4.897079703193594 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;rand \leq 3.8482795150864534 \cdot 10^{+31}:\\ \;\;\;\;a - 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	0.9
Cost	6976

\[a - \sqrt{a - 0.3333333333333333} \cdot \left(rand \cdot -0.3333333333333333\right) \]

Alternative 8
Error	18.6
Cost	192

\[a - 0.3333333333333333 \]

Alternative 9
Error	19.4
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022294 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))