Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B

Average Error: 0.2 → 0.2

Time: 32.1s

Precision: binary64

Cost: 26176

Math

\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]

↓

\[\mathsf{fma}\left(y \cdot \frac{\sqrt{z}}{\sqrt{2}}, \sqrt{0.5}, 0.5 \cdot x\right) \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

↓

(FPCore (x y z)
 :precision binary64
 (fma (* y (/ (sqrt z) (sqrt 2.0))) (sqrt 0.5) (* 0.5 x)))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

↓

double code(double x, double y, double z) {
	return fma((y * (sqrt(z) / sqrt(2.0))), sqrt(0.5), (0.5 * x));
}

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

↓

function code(x, y, z)
	return fma(Float64(y * Float64(sqrt(z) / sqrt(2.0))), sqrt(0.5), Float64(0.5 * x))
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(y * N[(N[Sqrt[z], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]

2. Applied egg-rr 0.5

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot \sqrt{z}\right) \cdot \sqrt{0.5}, \sqrt{0.5}, 0.5 \cdot x\right)} \]

3. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(\sqrt{z} \cdot \sqrt{0.5}\right), \sqrt{0.5}, 0.5 \cdot x\right)} \]
   Proof

4. Applied egg-rr 0.2

   \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\frac{\sqrt{z}}{\sqrt{2}}}, \sqrt{0.5}, 0.5 \cdot x\right) \]

Alternatives

Alternative 1
Error	0.2
Cost	13120

\[0.5 \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right) \]

Alternative 2
Error	16.2
Cost	7248

\[\begin{array}{l} t_0 := \left(0.5 \cdot y\right) \cdot \sqrt{z}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -13500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-73}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3
Error	0.2
Cost	6848

\[0.5 \cdot \left(\sqrt{z} \cdot y + x\right) \]

Alternative 4
Error	30.1
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))