HairBSDF, Mp, lower

Percentage Accurate: 99.8% → 99.9%

Time: 30.1s

Precision: binary32

Cost: 9888

• 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

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

↓

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

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
 (/ (* 0.5 (* (exp 0.6931) (pow E (/ -1.0 v)))) 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 (0.5f * (expf(0.6931f) * powf(((float) M_E), (-1.0f / v)))) / 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(Float32(0.5) * Float32(exp(Float32(0.6931)) * (Float32(exp(1)) ^ Float32(Float32(-1.0) / v)))) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

↓

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * (exp(single(0.6931)) * (single(2.71828182845904523536) ^ (single(-1.0) / v)))) / v;
end

Derivation

1. Initial program 99.5%

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

2. Simplified 99.5%

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

   [Start] 99.5%	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   exp-sum [=>] 99.5%	\[ \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Taylor expanded in cosTheta_i around 0 99.5%

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

4. Simplified 99.5%

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

   [Start] 99.5%	\[ e^{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   associate-*r/ [=>] 99.5%	\[ e^{\color{blue}{\frac{-1 \cdot \left(sinTheta_i \cdot sinTheta_O\right)}{v}} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   neg-mul-1 [<=] 99.5%	\[ e^{\frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   distribute-rgt-neg-in [=>] 99.5%	\[ e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]

5. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{0.6931 - \frac{1}{v}}}{v}} \]

6. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{0.5 \cdot e^{0.6931 + \frac{-1}{v}}}{v}} \]
   Step-by-step derivation

   [Start] 99.8%	\[ 0.5 \cdot \frac{e^{0.6931 - \frac{1}{v}}}{v} \]
   associate-*r/ [=>] 99.8%	\[ \color{blue}{\frac{0.5 \cdot e^{0.6931 - \frac{1}{v}}}{v}} \]
   sub-neg [=>] 99.8%	\[ \frac{0.5 \cdot e^{\color{blue}{0.6931 + \left(-\frac{1}{v}\right)}}}{v} \]
   distribute-neg-frac [=>] 99.8%	\[ \frac{0.5 \cdot e^{0.6931 + \color{blue}{\frac{-1}{v}}}}{v} \]
   metadata-eval [=>] 99.8%	\[ \frac{0.5 \cdot e^{0.6931 + \frac{\color{blue}{-1}}{v}}}{v} \]

7. Applied egg-rr 99.9%

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

   [Start] 99.8%	\[ \frac{0.5 \cdot e^{0.6931 + \frac{-1}{v}}}{v} \]
   exp-sum [=>] 99.9%	\[ \frac{0.5 \cdot \color{blue}{\left(e^{0.6931} \cdot e^{\frac{-1}{v}}\right)}}{v} \]

8. Applied egg-rr 99.9%

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

   [Start] 99.9%	\[ \frac{0.5 \cdot \left(e^{0.6931} \cdot e^{\frac{-1}{v}}\right)}{v} \]
   *-un-lft-identity [=>] 99.9%	\[ \frac{0.5 \cdot \left(e^{0.6931} \cdot e^{\color{blue}{1 \cdot \frac{-1}{v}}}\right)}{v} \]
   exp-prod [=>] 99.9%	\[ \frac{0.5 \cdot \left(e^{0.6931} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-1}{v}\right)}}\right)}{v} \]
   exp-1-e [=>] 99.9%	\[ \frac{0.5 \cdot \left(e^{0.6931} \cdot {\color{blue}{e}}^{\left(\frac{-1}{v}\right)}\right)}{v} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	9888

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

Alternative 2
Accuracy	99.9%
Cost	6688

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

Alternative 3
Accuracy	99.9%
Cost	3488

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

Alternative 4
Accuracy	99.4%
Cost	3424

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

Alternative 5
Accuracy	99.4%
Cost	3296

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

Alternative 6
Accuracy	4.3%
Cost	96

\[\frac{0.5}{v} \]

Reproduce

herbie shell --seed 2023256 
(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))))))