UniformSampleCone, z

Average Error: 0.11% → 0.05%

Time: 8.9s

Precision: binary32

Cost: 3360

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\left(1 - ux\right) + ux \cdot maxCos \]

↓

\[\mathsf{fma}\left(maxCos + -1, ux, 1\right) \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))

↓

(FPCore (ux uy maxCos) :precision binary32 (fma (+ maxCos -1.0) ux 1.0))

C

float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}

↓

float code(float ux, float uy, float maxCos) {
	return fmaf((maxCos + -1.0f), ux, 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end

↓

function code(ux, uy, maxCos)
	return fma(Float32(maxCos + Float32(-1.0)), ux, Float32(1.0))
end

Derivation

1. Initial program 0.11

   \[\left(1 - ux\right) + ux \cdot maxCos \]

2. Taylor expanded in ux around 0 0.06

   \[\leadsto \color{blue}{1 + \left(maxCos - 1\right) \cdot ux} \]

3. Simplified 0.05

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos + -1, ux, 1\right)} \]
   Proof

   [Start] 0.06	\[ 1 + \left(maxCos - 1\right) \cdot ux \]
   +-commutative [=>] 0.06	\[ \color{blue}{\left(maxCos - 1\right) \cdot ux + 1} \]
   fma-def [=>] 0.05	\[ \color{blue}{\mathsf{fma}\left(maxCos - 1, ux, 1\right)} \]
   sub-neg [=>] 0.05	\[ \mathsf{fma}\left(\color{blue}{maxCos + \left(-1\right)}, ux, 1\right) \]
   metadata-eval [=>] 0.05	\[ \mathsf{fma}\left(maxCos + \color{blue}{-1}, ux, 1\right) \]

4. Final simplification 0.05

   \[\leadsto \mathsf{fma}\left(maxCos + -1, ux, 1\right) \]

Alternatives

Alternative 1
Error	0.11%
Cost	224

\[\left(1 - ux\right) + maxCos \cdot ux \]

Alternative 2
Error	1.91%
Cost	96

\[1 - ux \]

Alternative 3
Error	28.51%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, z"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))