Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 0.7

Time: 12.9s

Precision: binary32

Cost: 6624

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

↓

\[s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right) \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

↓

(FPCore (s u)
 :precision binary32
 (* s (* (log1p (fma u -1.3333333333333333 0.3333333333333333)) -3.0)))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

↓

float code(float s, float u) {
	return s * (log1pf(fmaf(u, -1.3333333333333333f, 0.3333333333333333f)) * -3.0f);
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

↓

function code(s, u)
	return Float32(s * Float32(log1p(fma(u, Float32(-1.3333333333333333), Float32(0.3333333333333333))) * Float32(-3.0)))
end

Derivation

1. Initial program 1.3

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Simplified 0.7

   \[\leadsto \color{blue}{s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right)} \]
   Proof
   (*.f32 s (*.f32 (log1p.f32 (fma.f32 u -4/3 1/3)) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (fma.f32 u (Rewrite<= metadata-eval (-.f32 -1/3 1)) 1/3)) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (fma.f32 u (-.f32 (Rewrite<= metadata-eval (neg.f32 1/3)) 1) 1/3)) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (fma.f32 u (-.f32 (neg.f32 (Rewrite<= metadata-eval (/.f32 1/4 3/4))) 1) 1/3)) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (fma.f32 u (-.f32 (neg.f32 (/.f32 1/4 3/4)) 1) (Rewrite<= metadata-eval (/.f32 1/4 3/4)))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 u (-.f32 (neg.f32 (/.f32 1/4 3/4)) 1)) (/.f32 1/4 3/4)))) -3)): 77 points increase in error, 13 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (Rewrite=> *-commutative_binary32 (*.f32 (-.f32 (neg.f32 (/.f32 1/4 3/4)) 1) u)) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (*.f32 (-.f32 (neg.f32 (Rewrite=> metadata-eval 1/3)) 1) u) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (*.f32 (-.f32 (Rewrite=> metadata-eval -1/3) 1) u) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (*.f32 (Rewrite=> metadata-eval -4/3) u) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (*.f32 (Rewrite<= metadata-eval (/.f32 -1 3/4)) u) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (Rewrite<= associate-/r/_binary32 (/.f32 -1 (/.f32 3/4 u))) (/.f32 1/4 3/4))) -3)): 67 points increase in error, 5 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 -1 u) 3/4)) (/.f32 1/4 3/4))) -3)): 16 points increase in error, 32 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (/.f32 (Rewrite<= neg-mul-1_binary32 (neg.f32 u)) 3/4) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 u 3/4))) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (Rewrite=> neg-sub0_binary32 (-.f32 0 (/.f32 u 3/4))) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (+.f32 (-.f32 (Rewrite<= metadata-eval (log.f32 1)) (/.f32 u 3/4)) (/.f32 1/4 3/4))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (Rewrite<= associate--r-_binary32 (-.f32 (log.f32 1) (-.f32 (/.f32 u 3/4) (/.f32 1/4 3/4))))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (-.f32 (Rewrite=> metadata-eval 0) (-.f32 (/.f32 u 3/4) (/.f32 1/4 3/4)))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (-.f32 0 (Rewrite<= div-sub_binary32 (/.f32 (-.f32 u 1/4) 3/4)))) -3)): 12 points increase in error, 121 points decrease in error
   (*.f32 s (*.f32 (log1p.f32 (Rewrite<= neg-sub0_binary32 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4)))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4))))) -3)): 89 points increase in error, 7 points decrease in error
   (*.f32 s (*.f32 (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (log.f32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))) (Rewrite<= metadata-eval (neg.f32 3)))): 0 points increase in error, 0 points decrease in error
   (*.f32 s (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 (log.f32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))) 3)))): 0 points increase in error, 0 points decrease in error
   (*.f32 s (Rewrite<= distribute-lft-neg-out_binary32 (*.f32 (neg.f32 (log.f32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))) 3))): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))) 3)): 70 points increase in error, 41 points decrease in error
   (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 s (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))) 3)): 41 points increase in error, 53 points decrease in error
   (Rewrite<= *-commutative_binary32 (*.f32 3 (*.f32 s (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 3 s) (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))))): 57 points increase in error, 50 points decrease in error

3. Final simplification 0.7

   \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \cdot -3\right) \]

Alternatives

Alternative 1
Error	0.7
Cost	6624

\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right) \]

Alternative 2
Error	1.2
Cost	3488

\[-3 \cdot \left(s \cdot \log \left(u \cdot -1.3333333333333333 + 1.3333333333333333\right)\right) \]

Alternative 3
Error	1.0
Cost	3488

\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \]

Alternative 4
Error	1.0
Cost	3488

\[s \cdot \left(-3 \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \]

Alternative 5
Error	0.7
Cost	3488

\[\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u + -0.25\right)\right) \cdot \left(s \cdot -3\right) \]

Alternative 6
Error	22.5
Cost	224

\[-3 \cdot \left(-1.3333333333333333 \cdot \left(s \cdot u\right)\right) \]

Alternative 7
Error	22.5
Cost	160

\[u \cdot \left(s \cdot 4\right) \]

Alternative 8
Error	22.5
Cost	160

\[4 \cdot \left(s \cdot u\right) \]

Reproduce

herbie shell --seed 2022294 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))