Average Error: 6.0 → 0.1
Time: 6.6s
Precision: binary64
Cost: 6720
\[x + \frac{y \cdot y}{z} \]
↓
\[\mathsf{fma}\left(y, \frac{y}{z}, x\right) \]
(FPCore (x y z) :precision binary64 (+ x (/ (* y y) z)))
↓
(FPCore (x y z) :precision binary64 (fma y (/ y z) x))
double code(double x, double y, double z) {
	return x + ((y * y) / z);
}
↓
double code(double x, double y, double z) {
	return fma(y, (y / z), x);
}

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

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

x + \frac{y \cdot y}{z}
↓
\mathsf{fma}\left(y, \frac{y}{z}, x\right)

Error

Target

Original6.0
Target0.1
Herbie0.1
\[x + y \cdot \frac{y}{z} \]

Derivation

  1. Initial program 6.0

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{y}{z}, x\right)} \]
    Proof
    (fma.f64 y (/.f64 y z) x): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (/.f64 y z)) x)): 3 points increase in error, 1 points decrease in error
    (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y y) z)) x): 36 points increase in error, 15 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y y) z))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error11.5
Cost1872
\[\begin{array}{l} t_0 := \frac{y \cdot y}{z}\\ t_1 := \frac{y}{\frac{z}{y}}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error14.6
Cost584
\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{y}}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error0.1
Cost448
\[x + y \cdot \frac{y}{z} \]
Alternative 4
Error21.0
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022294 
(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)))