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

↓

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

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

↓

(FPCore (x y z) :precision binary64 (fma (/ y z) y 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 / z), y, x);
}

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

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]

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

↓

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

Original	6.2
Target	0.1
Herbie	0.1

Derivation

Initial program 6.2
\[x + \frac{y \cdot y}{z} \]
Simplified0.1
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, y, x\right)} \]
Proof
(fma.f64 (/.f64 y z) y x): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 y z) y) x)): 3 points increase in error, 0 points decrease in error
(+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y y) z)) x): 45 points increase in error, 16 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 y y) z))): 0 points increase in error, 0 points decrease in error
Final simplification0.1
\[\leadsto \mathsf{fma}\left(\frac{y}{z}, y, x\right) \]

Alternatives

Alternative 1
Error	12.0
Cost	1872

\[\begin{array}{l} t_0 := y \cdot \frac{y}{z}\\ t_1 := \frac{y \cdot y}{z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\frac{y}{\frac{z}{y}}\\ \mathbf{elif}\;t_1 \leq -4000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-91}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	14.8
Cost	584

\[\begin{array}{l} t_0 := y \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	448

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

Alternative 4
Error	0.1
Cost	448

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

Alternative 5
Error	21.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))

Error

Target

Derivation

Alternatives

Error

Reproduce