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

Average Error: 0.02% → 0.01%

Time: 3.7s

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	29.57%
Cost	849

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+84}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{+16} \lor \neg \left(y \leq -1.56 \cdot 10^{-24}\right) \land y \leq 1.55 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 2
Error	39.61%
Cost	652

\[\begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-51}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+181}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \end{array} \]

Alternative 3
Error	21.77%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-51} \lor \neg \left(z \leq 2.7 \cdot 10^{-117}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 4
Error	1.91%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 8.6 \cdot 10^{-13}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 5
Error	39.91%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-51}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 6
Error	0.02%
Cost	448

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

Alternative 7
Error	55.45%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))