HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 17.6s

Precision: binary32

Cost: 16416

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

↓

\[\frac{\frac{e^{0.6931 + \left(\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v} + \mathsf{log1p}\left(-0.5\right)\right)}}{\sqrt{v}}}{\sqrt{v}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/
   (exp (+ 0.6931 (+ (/ (fma cosTheta_i cosTheta_O -1.0) v) (log1p -0.5))))
   (sqrt v))
  (sqrt v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (2.0f * v)))));
}

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf((0.6931f + ((fmaf(cosTheta_i, cosTheta_O, -1.0f) / v) + log1pf(-0.5f)))) / sqrtf(v)) / sqrtf(v);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

↓

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(Float32(0.6931) + Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v) + log1p(Float32(-0.5))))) / sqrt(v)) / sqrt(v))
end

Derivation

1. Initial program 0.1

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
   Proof
   (exp.f32 (+.f32 (-.f32 (/.f32 cosTheta_i (/.f32 v cosTheta_O)) (+.f32 (/.f32 sinTheta_i (/.f32 v sinTheta_O)) (/.f32 1 v))) (+.f32 6931/10000 (log.f32 (/.f32 1/2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (+.f32 (/.f32 sinTheta_i (/.f32 v sinTheta_O)) (/.f32 1 v))) (+.f32 6931/10000 (log.f32 (/.f32 1/2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (+.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v))) (+.f32 6931/10000 (log.f32 (/.f32 1/2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (+.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v) (Rewrite<= remove-double-neg_binary32 (neg.f32 (neg.f32 (/.f32 1 v)))))) (+.f32 6931/10000 (log.f32 (/.f32 1/2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (Rewrite<= sub-neg_binary32 (-.f32 (/.f32 (*.f32 sinTheta_i sinTheta_O) v) (neg.f32 (/.f32 1 v))))) (+.f32 6931/10000 (log.f32 (/.f32 1/2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (Rewrite<= associate-+l-_binary32 (+.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (neg.f32 (/.f32 1 v)))) (+.f32 6931/10000 (log.f32 (/.f32 1/2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (Rewrite<= sub-neg_binary32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v))) (+.f32 6931/10000 (log.f32 (/.f32 1/2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) (+.f32 6931/10000 (log.f32 (/.f32 (Rewrite<= metadata-eval (/.f32 1 2)) v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) (+.f32 6931/10000 (log.f32 (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 2 v))))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) (+.f32 6931/10000 (Rewrite=> log-rec_binary32 (neg.f32 (log.f32 (*.f32 2 v))))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) (Rewrite=> unsub-neg_binary32 (-.f32 6931/10000 (log.f32 (*.f32 2 v)))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000) (log.f32 (*.f32 2 v))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (Rewrite<= unsub-neg_binary32 (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000) (neg.f32 (log.f32 (*.f32 2 v)))))): 0 points increase in error, 0 points decrease in error
   (exp.f32 (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 1 v)) 6931/10000) (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (*.f32 2 v)))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

   \[\leadsto e^{\color{blue}{\left(\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931\right) + \log 0.5\right) - \log v}} \]

4. Taylor expanded in sinTheta_i around 0 0.1

   \[\leadsto \color{blue}{e^{\left(0.6931 + \left(\log 0.5 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \left(\frac{1}{v} + \log v\right)}} \]

5. Simplified 0.1

   \[\leadsto \color{blue}{\frac{e^{\left(0.6931 + \log 0.5\right) + \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \frac{-1}{v}\right)}}{v}} \]
   Proof
   (/.f32 (exp.f32 (+.f32 (+.f32 6931/10000 (log.f32 1/2)) (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v)))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (+.f32 (+.f32 6931/10000 (log.f32 1/2)) (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 (Rewrite<= metadata-eval (neg.f32 1)) v)))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (+.f32 (+.f32 6931/10000 (log.f32 1/2)) (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 1 v)))))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (+.f32 (+.f32 6931/10000 (log.f32 1/2)) (Rewrite<= fma-neg_binary32 (-.f32 (*.f32 cosTheta_O (/.f32 cosTheta_i v)) (/.f32 1 v))))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (+.f32 (+.f32 6931/10000 (log.f32 1/2)) (-.f32 (Rewrite<= *-commutative_binary32 (*.f32 (/.f32 cosTheta_i v) cosTheta_O)) (/.f32 1 v)))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (+.f32 (+.f32 6931/10000 (log.f32 1/2)) (-.f32 (Rewrite=> associate-*l/_binary32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 1 v)))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (Rewrite<= associate--l+_binary32 (-.f32 (+.f32 (+.f32 6931/10000 (log.f32 1/2)) (/.f32 (*.f32 cosTheta_i cosTheta_O) v)) (/.f32 1 v)))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (-.f32 (Rewrite<= associate-+r+_binary32 (+.f32 6931/10000 (+.f32 (log.f32 1/2) (/.f32 (*.f32 cosTheta_i cosTheta_O) v)))) (/.f32 1 v))) v): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (-.f32 (+.f32 6931/10000 (+.f32 (log.f32 1/2) (/.f32 (*.f32 cosTheta_i cosTheta_O) v))) (/.f32 1 v))) (Rewrite<= unpow1_binary32 (pow.f32 v 1))): 0 points increase in error, 0 points decrease in error
   (/.f32 (exp.f32 (-.f32 (+.f32 6931/10000 (+.f32 (log.f32 1/2) (/.f32 (*.f32 cosTheta_i cosTheta_O) v))) (/.f32 1 v))) (Rewrite<= exp-to-pow_binary32 (exp.f32 (*.f32 (log.f32 v) 1)))): 1 points increase in error, 1 points decrease in error
   (/.f32 (exp.f32 (-.f32 (+.f32 6931/10000 (+.f32 (log.f32 1/2) (/.f32 (*.f32 cosTheta_i cosTheta_O) v))) (/.f32 1 v))) (exp.f32 (Rewrite=> *-rgt-identity_binary32 (log.f32 v)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= exp-diff_binary32 (exp.f32 (-.f32 (-.f32 (+.f32 6931/10000 (+.f32 (log.f32 1/2) (/.f32 (*.f32 cosTheta_i cosTheta_O) v))) (/.f32 1 v)) (log.f32 v)))): 2 points increase in error, 3 points decrease in error
   (exp.f32 (Rewrite<= associate--r+_binary32 (-.f32 (+.f32 6931/10000 (+.f32 (log.f32 1/2) (/.f32 (*.f32 cosTheta_i cosTheta_O) v))) (+.f32 (/.f32 1 v) (log.f32 v))))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.1

   \[\leadsto \frac{\color{blue}{{\left(e^{1}\right)}^{\left(0.6931 + \left(\log 0.5 + \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \frac{-1}{v}\right)\right)\right)}}}{v} \]

7. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{1}{\sqrt{v}} \cdot \frac{e^{\mathsf{log1p}\left(-0.5\right) + \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \frac{-1}{v}\right) + 0.6931\right)}}{\sqrt{v}}} \]

8. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\frac{e^{0.6931 + \left(\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v} + \mathsf{log1p}\left(-0.5\right)\right)}}{\sqrt{v}}}{\sqrt{v}}} \]
   Proof
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (/.f32 (fma.f32 cosTheta_i cosTheta_O -1) v) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (/.f32 (fma.f32 cosTheta_i cosTheta_O (Rewrite<= metadata-eval (neg.f32 1))) v) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (/.f32 (Rewrite<= fma-neg_binary32 (-.f32 (*.f32 cosTheta_i cosTheta_O) 1)) v) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 1 v))) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (-.f32 (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 cosTheta_O cosTheta_i)) v) (/.f32 1 v)) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (-.f32 (Rewrite<= associate-*r/_binary32 (*.f32 cosTheta_O (/.f32 cosTheta_i v))) (/.f32 1 v)) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (Rewrite=> fma-neg_binary32 (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (neg.f32 (/.f32 1 v)))) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (Rewrite=> distribute-neg-frac_binary32 (/.f32 (neg.f32 1) v))) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (+.f32 (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 (Rewrite=> metadata-eval -1) v)) (log1p.f32 -1/2)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 6931/10000 (Rewrite<= +-commutative_binary32 (+.f32 (log1p.f32 -1/2) (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v)))))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 6931/10000 (log1p.f32 -1/2)) (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v))))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 (Rewrite=> +-commutative_binary32 (+.f32 (log1p.f32 -1/2) 6931/10000)) (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (Rewrite=> associate-+l+_binary32 (+.f32 (log1p.f32 -1/2) (+.f32 6931/10000 (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v)))))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (exp.f32 (+.f32 (log1p.f32 -1/2) (Rewrite<= +-commutative_binary32 (+.f32 (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v)) 6931/10000)))) (sqrt.f32 v)) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 (/.f32 (exp.f32 (+.f32 (log1p.f32 -1/2) (+.f32 (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v)) 6931/10000))) (sqrt.f32 v)))) (sqrt.f32 v)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 1 (sqrt.f32 v)) (/.f32 (exp.f32 (+.f32 (log1p.f32 -1/2) (+.f32 (fma.f32 cosTheta_O (/.f32 cosTheta_i v) (/.f32 -1 v)) 6931/10000))) (sqrt.f32 v)))): 0 points increase in error, 1 points decrease in error

9. Final simplification 0.1

   \[\leadsto \frac{\frac{e^{0.6931 + \left(\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v} + \mathsf{log1p}\left(-0.5\right)\right)}}{\sqrt{v}}}{\sqrt{v}} \]

Alternatives

Alternative 1
Error	0.1
Cost	13248

\[{\left(\frac{v}{e^{\mathsf{log1p}\left(-0.5\right) + \left(0.6931 + \mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \frac{-1}{v}\right)\right)}}\right)}^{-1} \]

Alternative 2
Error	0.1
Cost	6688

\[\frac{e^{\left(0.6931 + \log 0.5\right) + \frac{-1}{v}}}{v} \]

Alternative 3
Error	0.1
Cost	3488

\[e^{0.6931 + \frac{-1}{v}} \cdot \frac{0.5}{v} \]

Alternative 4
Error	0.3
Cost	3360

\[\frac{e^{\frac{-1}{v}}}{v} \]

Alternative 5
Error	29.9
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022330 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))