?

Average Error: 0.2% → 0.19%
Time: 10.9s
Precision: binary64
Cost: 19520

?

\[x \cdot \sin y + z \cdot \cos y \]
↓
\[\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right) \]
(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))
↓
(FPCore (x y z) :precision binary64 (fma x (sin y) (* z (cos y))))
double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(y));
}
↓
double code(double x, double y, double z) {
	return fma(x, sin(y), (z * cos(y)));
}

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

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

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

Error?

Derivation?

  1. Initial program 0.2

    \[x \cdot \sin y + z \cdot \cos y \]

  2. Simplified0.19

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

    [Start]0.2

    \[ x \cdot \sin y + z \cdot \cos y \]

    fma-def [=>]0.19

    \[ \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)} \]

  3. Final simplification0.19

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

Alternatives

Alternative 1
Error0.2%
Cost13248
\[z \cdot \cos y + x \cdot \sin y \]
Alternative 2
Error25.96%
Cost7517
\[\begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.0015:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.00028:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+162} \lor \neg \left(y \leq 1.3 \cdot 10^{+235}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error16.03%
Cost7250
\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+83} \lor \neg \left(z \leq 2.75 \cdot 10^{-98} \lor \neg \left(z \leq 8.6 \cdot 10^{-6}\right) \land z \leq 2.15 \cdot 10^{+98}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot \sin y\\ \end{array} \]
Alternative 4
Error26.2%
Cost6857
\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \lor \neg \left(y \leq 1.5 \cdot 10^{-5}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]
Alternative 5
Error60.46%
Cost588
\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+197}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+290}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
Alternative 6
Error49.11%
Cost320
\[z + x \cdot y \]
Alternative 7
Error62.12%
Cost64
\[z \]

Error

Reproduce?

herbie shell --seed 2023102 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))