HairBSDF, Mp, lower

Percentage Accurate: 49.7% → 99.3%

Time: 26.4s

Precision: binary32

Cost: 12960

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

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)\right) \]

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
 (log1p (expm1 (exp (/ (fma cosTheta_i cosTheta_O -1.0) 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 log1pf(expm1f(expf((fmaf(cosTheta_i, cosTheta_O, -1.0f) / 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 log1p(expm1(exp(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v))))
end

Derivation

1. Initial program 46.9%

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

2. Simplified 46.9%

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

   [Start] 46.9%	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   associate-+l+ [=>] 46.9%	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]
   sub-neg [=>] 46.9%	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [=>] 46.9%	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [<=] 46.9%	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   sub-neg [<=] 46.9%	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate--l- [=>] 46.9%	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/l* [=>] 46.9%	\[ e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/r* [=>] 46.9%	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]
   metadata-eval [=>] 46.9%	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Taylor expanded in v around 0 99.3%

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

4. Taylor expanded in sinTheta_i around 0 99.3%

   \[\leadsto \color{blue}{e^{\frac{cosTheta_i \cdot cosTheta_O - 1}{v}}} \]

5. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)\right)} \]
   Step-by-step derivation

   [Start] 99.3%	\[ e^{\frac{cosTheta_i \cdot cosTheta_O - 1}{v}} \]
   log1p-expm1-u [=>] 99.7%	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{cosTheta_i \cdot cosTheta_O - 1}{v}}\right)\right)} \]
   fma-neg [=>] 99.7%	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{\color{blue}{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}}{v}}\right)\right) \]
   metadata-eval [=>] 99.7%	\[ \mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, \color{blue}{-1}\right)}{v}}\right)\right) \]

6. Final simplification 99.7%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	12960

\[\mathsf{log1p}\left(\mathsf{expm1}\left(e^{\frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)\right) \]

Alternative 2
Accuracy	98.7%
Cost	3296

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

Alternative 3
Accuracy	8.1%
Cost	352

\[\left(1 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) + \frac{-1}{v} \]

Alternative 4
Accuracy	6.3%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023255 
(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))))))