Average Error: 0.1 → 0.1
Time: 7.5s
Precision: binary64
Cost: 19520
\[x \cdot \sin y + z \cdot \cos y \]
↓
\[\mathsf{fma}\left(\cos y, z, \sin y \cdot x\right) \]
(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))
↓
(FPCore (x y z) :precision binary64 (fma (cos y) z (* (sin y) x)))
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(cos(y), z, (sin(y) * x));
}

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end
↓
function code(x, y, z)
	return fma(cos(y), z, Float64(sin(y) * x))
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[(N[Cos[y], $MachinePrecision] * z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{\cos y \cdot z + \sin y \cdot x} \]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, \sin y \cdot x\right)} \]
    Proof
    (fma.f64 (cos.f64 y) z (*.f64 (sin.f64 y) x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 y) z) (*.f64 (sin.f64 y) x))): 1 points increase in error, 0 points decrease in error

  4. Final simplification0.1

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

Alternatives

Alternative 1
Error0.1
Cost13248
\[\sin y \cdot x + \cos y \cdot z \]
Alternative 2
Error16.8
Cost7120
\[\begin{array}{l} t_0 := \sin y \cdot x\\ \mathbf{if}\;x \leq -4 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+173}:\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error9.1
Cost6984
\[\begin{array}{l} t_0 := z + \sin y \cdot x\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+23}:\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error16.4
Cost6856
\[\begin{array}{l} t_0 := \cos y \cdot z\\ \mathbf{if}\;y \leq -0.00155:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;z + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error37.1
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+151}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+86}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
Alternative 6
Error30.4
Cost320
\[z + y \cdot x \]
Alternative 7
Error38.4
Cost64
\[z \]

Error

Reproduce

herbie shell --seed 2022338 
(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))))