Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.0% → 96.0%

Time: 6.9s

Precision: binary64

Cost: 6720

Math

\[\frac{x \cdot \left(y + z\right)}{z} \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	84.0%
Target	96.3%
Herbie	96.0%

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

Derivation

1. Initial program 85.9%

   \[\frac{x \cdot \left(y + z\right)}{z} \]

2. Simplified 97.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]
   Step-by-step derivation

   [Start] 85.9%	\[ \frac{x \cdot \left(y + z\right)}{z} \]
   associate-*l/ [<=] 88.2%	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
   distribute-rgt-in [=>] 82.7%	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
   *-commutative [=>] 82.7%	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
   associate-/r/ [<=] 95.1%	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
   *-inverses [=>] 95.1%	\[ y \cdot \frac{x}{z} + \frac{x}{\color{blue}{1}} \]
   /-rgt-identity [=>] 95.1%	\[ y \cdot \frac{x}{z} + \color{blue}{x} \]
   associate-*r/ [=>] 95.8%	\[ \color{blue}{\frac{y \cdot x}{z}} + x \]
   *-commutative [<=] 95.8%	\[ \frac{\color{blue}{x \cdot y}}{z} + x \]
   associate-*r/ [<=] 97.7%	\[ \color{blue}{x \cdot \frac{y}{z}} + x \]
   fma-def [=>] 97.7%	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]

3. Final simplification 97.7%

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

Alternatives

Alternative 1
Accuracy	96.0%
Cost	6720

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

Alternative 2
Accuracy	89.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+248}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+146}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	71.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -24000000000 \lor \neg \left(y \leq 2.55 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	73.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -22000000000 \lor \neg \left(y \leq 1.22 \cdot 10^{-33}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	96.0%
Cost	448

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

Alternative 6
Accuracy	96.3%
Cost	448

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

Alternative 7
Accuracy	50.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

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

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