Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B

Average Error: 0.02% → 0.01%

Time: 5.4s

Precision: binary64

Cost: 6720

Math

\[x + \left(y - x\right) \cdot z \]

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma (- y x) z x))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

↓

double code(double x, double y, double z) {
	return fma((y - x), z, x);
}

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - x) * z))
end

↓

function code(x, y, z)
	return fma(Float64(y - x), z, x)
end

Wolfram

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

↓

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

Derivation

1. Initial program 0.02

   \[x + \left(y - x\right) \cdot z \]

2. Simplified 0.01

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

   [Start] 0.02	\[ x + \left(y - x\right) \cdot z \]
   +-commutative [=>] 0.02	\[ \color{blue}{\left(y - x\right) \cdot z + x} \]
   fma-def [=>] 0.01	\[ \color{blue}{\mathsf{fma}\left(y - x, z, x\right)} \]

3. Final simplification 0.01

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

Alternatives

Alternative 1
Error	37.57%
Cost	784

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+86}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.21%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -72000000 \lor \neg \left(z \leq 50\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 3
Error	1.52%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.0125\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 4
Error	27.23%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+83}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 5
Error	0.02%
Cost	576

\[x + \left(y \cdot z - x \cdot z\right) \]

Alternative 6
Error	36.78%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-8}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 7
Error	0.02%
Cost	448

\[x + \left(y - x\right) \cdot z \]

Alternative 8
Error	53.64%
Cost	64

\[x \]

Reproduce

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