Crypto.Random.Test:calculate from crypto-random-0.0.9

Average Error: 6.2 → 0.1

Time: 4.5s

Precision: binary64

Cost: 6720

Math

\[x + \frac{y \cdot y}{z} \]

↓

\[\mathsf{fma}\left(\frac{y}{z}, y, x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ (* y y) z)))

↓

(FPCore (x y z) :precision binary64 (fma (/ y z) y x))

C

double code(double x, double y, double z) {
	return x + ((y * y) / z);
}

↓

double code(double x, double y, double z) {
	return fma((y / z), y, x);
}

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * y) / z))
end

↓

function code(x, y, z)
	return fma(Float64(y / z), y, x)
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(y / z), $MachinePrecision] * y + x), $MachinePrecision]

Target

Original	6.2
Target	0.1
Herbie	0.1

\[x + y \cdot \frac{y}{z} \]

Derivation

1. Initial program 6.2

   \[x + \frac{y \cdot y}{z} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, y, x\right)} \]
   Proof

   [Start] 6.2	\[ x + \frac{y \cdot y}{z} \]
   +-commutative [=>] 6.2	\[ \color{blue}{\frac{y \cdot y}{z} + x} \]
   associate-*l/ [<=] 0.1	\[ \color{blue}{\frac{y}{z} \cdot y} + x \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(\frac{y}{z}, y, x\right)} \]

3. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(\frac{y}{z}, y, x\right) \]

Alternatives

Alternative 1
Error	10.8
Cost	1097

\[\begin{array}{l} t_0 := \frac{y \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+61} \lor \neg \left(t_0 \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;y \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

\[x + y \cdot \frac{y}{z} \]

Alternative 3
Error	20.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x y z)
  :name "Crypto.Random.Test:calculate from crypto-random-0.0.9"
  :precision binary64

  :herbie-target
  (+ x (* y (/ y z)))

  (+ x (/ (* y y) z)))