HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 21.4s

Precision: binary32

Cost: 19744

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

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

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
  (sqrt
   (*
    (*
     (pow
      (exp (- (* cosTheta_i cosTheta_O) (fma sinTheta_i sinTheta_O 1.0)))
      (/ 1.0 v))
     (exp 0.6931))
    (/ 0.5 v)))
  2.0))

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(sqrtf(((powf(expf(((cosTheta_i * cosTheta_O) - fmaf(sinTheta_i, sinTheta_O, 1.0f))), (1.0f / v)) * expf(0.6931f)) * (0.5f / v))), 2.0f);
}

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 sqrt(Float32(Float32((exp(Float32(Float32(cosTheta_i * cosTheta_O) - fma(sinTheta_i, sinTheta_O, Float32(1.0)))) ^ Float32(Float32(1.0) / v)) * exp(Float32(0.6931))) * Float32(Float32(0.5) / v))) ^ Float32(2.0)
end

Derivation

1. Initial program 99.7%

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

2. Simplified 99.7%

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

   [Start] 99.7	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   associate-+l+ [=>] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]
   sub-neg [=>] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [=>] 99.7	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [<=] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   sub-neg [<=] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate--l- [=>] 99.7	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/l* [=>] 99.7	\[ e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/r* [=>] 99.7	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]
   metadata-eval [=>] 99.7	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{\left(\sqrt{e^{\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v} \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)\right) + 0.6931} \cdot \frac{0.5}{v}}\right)}^{2}} \]
   Step-by-step derivation

   [Start] 99.7	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]
   add-sqr-sqrt [=>] 99.7	\[ \color{blue}{\sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]
   pow2 [=>] 99.7	\[ \color{blue}{{\left(\sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2}} \]

4. Applied egg-rr 99.8%

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

   [Start] 99.7	\[ {\left(\sqrt{e^{\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v} \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)\right) + 0.6931} \cdot \frac{0.5}{v}}\right)}^{2} \]
   exp-sum [=>] 99.8	\[ {\left(\sqrt{\color{blue}{\left(e^{cosTheta_i \cdot \frac{cosTheta_O}{v} - \frac{1}{v} \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)} \cdot e^{0.6931}\right)} \cdot \frac{0.5}{v}}\right)}^{2} \]
   associate-*r/ [=>] 99.8	\[ {\left(\sqrt{\left(e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} - \frac{1}{v} \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]
   associate-*l/ [=>] 99.8	\[ {\left(\sqrt{\left(e^{\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{1 \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)}{v}}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]
   *-un-lft-identity [<=] 99.8	\[ {\left(\sqrt{\left(e^{\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{\color{blue}{1 + sinTheta_i \cdot sinTheta_O}}{v}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]
   sub-div [=>] 99.8	\[ {\left(\sqrt{\left(e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - \left(1 + sinTheta_i \cdot sinTheta_O\right)}{v}}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]
   +-commutative [=>] 99.8	\[ {\left(\sqrt{\left(e^{\frac{cosTheta_i \cdot cosTheta_O - \color{blue}{\left(sinTheta_i \cdot sinTheta_O + 1\right)}}{v}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]
   fma-def [=>] 99.8	\[ {\left(\sqrt{\left(e^{\frac{cosTheta_i \cdot cosTheta_O - \color{blue}{\mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}}{v}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]

5. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ {\left(\sqrt{\left(e^{\frac{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}{v}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]
   div-inv [=>] 99.8	\[ {\left(\sqrt{\left(e^{\color{blue}{\left(cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)\right) \cdot \frac{1}{v}}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]
   exp-prod [=>] 99.8	\[ {\left(\sqrt{\left(\color{blue}{{\left(e^{cosTheta_i \cdot cosTheta_O - \mathsf{fma}\left(sinTheta_i, sinTheta_O, 1\right)}\right)}^{\left(\frac{1}{v}\right)}} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v}}\right)}^{2} \]

6. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7008

\[\frac{0.5}{v} \cdot {e}^{\left(\left(0.6931 + \frac{-1}{v}\right) - \frac{sinTheta_i \cdot sinTheta_O - cosTheta_i \cdot cosTheta_O}{v}\right)} \]

Alternative 2
Accuracy	99.7%
Cost	6880

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

Alternative 3
Accuracy	99.7%
Cost	6880

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

Alternative 4
Accuracy	99.7%
Cost	3616

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

Alternative 5
Accuracy	99.7%
Cost	3488

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

Alternative 6
Accuracy	98.1%
Cost	3424

\[e^{\frac{cosTheta_i \cdot cosTheta_O + -1}{v}} \]

Alternative 7
Accuracy	98.1%
Cost	3296

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

Alternative 8
Accuracy	20.1%
Cost	160

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

Alternative 9
Accuracy	38.8%
Cost	160

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

Alternative 10
Accuracy	6.4%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023160 
(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))))))