HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 18.2s

Precision: binary32

Cost: 23584

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

Math

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

↓

\[\left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left(\sqrt[3]{\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)}}{v}} \cdot \sqrt[3]{0.5}\right) \]

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
 (*
  (*
   (pow (/ 0.5 v) 0.6666666666666666)
   (pow
    (exp
     (+
      (-
       (* cosTheta_O (/ cosTheta_i v))
       (fma sinTheta_i (/ sinTheta_O v) (/ 1.0 v)))
      0.6931))
    0.6666666666666666))
  (*
   (cbrt (/ (exp (- 0.6931 (+ (/ 1.0 v) (/ sinTheta_i (/ v sinTheta_O))))) v))
   (cbrt 0.5))))

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 (powf((0.5f / v), 0.6666666666666666f) * powf(expf((((cosTheta_O * (cosTheta_i / v)) - fmaf(sinTheta_i, (sinTheta_O / v), (1.0f / v))) + 0.6931f)), 0.6666666666666666f)) * (cbrtf((expf((0.6931f - ((1.0f / v) + (sinTheta_i / (v / sinTheta_O))))) / v)) * cbrtf(0.5f));
}

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((Float32(Float32(0.5) / v) ^ Float32(0.6666666666666666)) * (exp(Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - fma(sinTheta_i, Float32(sinTheta_O / v), Float32(Float32(1.0) / v))) + Float32(0.6931))) ^ Float32(0.6666666666666666))) * Float32(cbrt(Float32(exp(Float32(Float32(0.6931) - Float32(Float32(Float32(1.0) / v) + Float32(sinTheta_i / Float32(v / sinTheta_O))))) / v)) * cbrt(Float32(0.5))))
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

   [Start] 0.1	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [=>] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   log-div [=>] 0.1	\[ e^{\color{blue}{\left(\log 1 - \log \left(2 \cdot v\right)\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   metadata-eval [=>] 0.1	\[ e^{\left(\color{blue}{0} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   associate-+l- [=>] 0.1	\[ e^{\color{blue}{0 - \left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   associate-+l- [<=] 0.1	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   metadata-eval [<=] 0.1	\[ e^{\left(\color{blue}{\log 1} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   log-div [<=] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   +-commutative [<=] 0.1	\[ e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]
   associate-+l+ [=>] 0.1	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

3. Applied egg-rr 0.4

   \[\leadsto \color{blue}{{\left({\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \cdot \frac{0.5}{v}\right)}^{3}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{{\left({\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \cdot \frac{0.5}{v}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \cdot \frac{0.5}{v}}} \]

5. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right)} \cdot \sqrt[3]{e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931} \cdot \frac{0.5}{v}} \]

6. Taylor expanded in cosTheta_O around 0 0.1

   \[\leadsto \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \color{blue}{\left({\left(\frac{1 \cdot e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.3333333333333333} \cdot \sqrt[3]{0.5}\right)} \]

7. Simplified 0.1

   \[\leadsto \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \color{blue}{\left(\sqrt[3]{\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)}}{v}} \cdot \sqrt[3]{0.5}\right)} \]
   Proof

   [Start] 0.1	\[ \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left({\left(\frac{1 \cdot e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}\right)}^{0.3333333333333333} \cdot \sqrt[3]{0.5}\right) \]
   unpow1/3 [=>] 0.1	\[ \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left(\color{blue}{\sqrt[3]{\frac{1 \cdot e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}{v}}} \cdot \sqrt[3]{0.5}\right) \]
   *-lft-identity [=>] 0.1	\[ \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left(\sqrt[3]{\frac{\color{blue}{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}}{v}} \cdot \sqrt[3]{0.5}\right) \]
   +-commutative [=>] 0.1	\[ \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left(\sqrt[3]{\frac{e^{0.6931 - \color{blue}{\left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)}}}{v}} \cdot \sqrt[3]{0.5}\right) \]
   +-commutative [<=] 0.1	\[ \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left(\sqrt[3]{\frac{e^{0.6931 - \color{blue}{\left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}}{v}} \cdot \sqrt[3]{0.5}\right) \]
   associate-/l* [=>] 0.1	\[ \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left(\sqrt[3]{\frac{e^{0.6931 - \left(\frac{1}{v} + \color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\right)}}{v}} \cdot \sqrt[3]{0.5}\right) \]

8. Final simplification 0.1

   \[\leadsto \left({\left(\frac{0.5}{v}\right)}^{0.6666666666666666} \cdot {\left(e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + 0.6931}\right)}^{0.6666666666666666}\right) \cdot \left(\sqrt[3]{\frac{e^{0.6931 - \left(\frac{1}{v} + \frac{sinTheta_i}{\frac{v}{sinTheta_O}}\right)}}{v}} \cdot \sqrt[3]{0.5}\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	13184

\[\frac{0.5}{v} \cdot \sqrt{{\left(e^{2}\right)}^{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}\right)}} \]

Alternative 2
Error	0.1
Cost	6688

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

Alternative 3
Error	0.1
Cost	3488

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

Alternative 4
Error	0.6
Cost	3424

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

Alternative 5
Error	0.7
Cost	3296

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

Alternative 6
Error	25.5
Cost	160

\[cosTheta_i \cdot \frac{cosTheta_O}{v} \]

Alternative 7
Error	25.5
Cost	160

\[cosTheta_O \cdot \frac{cosTheta_i}{v} \]

Alternative 8
Error	19.4
Cost	160

\[\frac{cosTheta_O \cdot cosTheta_i}{v} \]

Alternative 9
Error	29.9
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023046 
(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))))))