UniformSampleCone, z

Average Error: 0.0 → 0.0

Time: 7.0s

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 0.0

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

2. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\frac{1 - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{3}}{1 + \left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(1 - maxCos\right)\right)}} \]

3. Simplified 0.4

   \[\leadsto \color{blue}{\frac{1 - {\left(ux \cdot \left(1 - maxCos\right)\right)}^{3}}{1 + \left(1 + ux \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}} \]
   Proof
   (/.f32 (-.f32 1 (pow.f32 (*.f32 ux (-.f32 1 maxCos)) 3)) (+.f32 1 (*.f32 (+.f32 1 (*.f32 ux (-.f32 1 maxCos))) (*.f32 ux (-.f32 1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (-.f32 1 (pow.f32 (*.f32 ux (-.f32 1 maxCos)) 3)) (+.f32 1 (*.f32 (Rewrite=> +-commutative_binary32 (+.f32 (*.f32 ux (-.f32 1 maxCos)) 1)) (*.f32 ux (-.f32 1 maxCos))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (-.f32 1 (pow.f32 (*.f32 ux (-.f32 1 maxCos)) 3)) (+.f32 1 (Rewrite<= distribute-lft1-in_binary32 (+.f32 (*.f32 (*.f32 ux (-.f32 1 maxCos)) (*.f32 ux (-.f32 1 maxCos))) (*.f32 ux (-.f32 1 maxCos)))))): 1 points increase in error, 1 points decrease in error

4. Taylor expanded in ux around 0 0.0

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

5. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos + -1, ux, 1\right)} \]
   Proof
   (fma.f32 (+.f32 maxCos -1) ux 1): 0 points increase in error, 0 points decrease in error
   (fma.f32 (+.f32 maxCos (Rewrite<= metadata-eval (neg.f32 1))) ux 1): 0 points increase in error, 0 points decrease in error
   (fma.f32 (Rewrite<= sub-neg_binary32 (-.f32 maxCos 1)) ux 1): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (-.f32 maxCos 1) ux) 1)): 5 points increase in error, 1 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 (-.f32 maxCos 1) ux))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	224

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

Alternative 2
Error	0.0
Cost	224

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

Alternative 3
Error	0.6
Cost	96

\[1 - ux \]

Alternative 4
Error	9.2
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022329 
(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)))