HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 16.7s

Precision: binary32

Cost: 16448

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

Math

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

↓

\[{\left({\left({\left(e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)}^{0.16666666666666666}\right)}^{2}\right)}^{3} \cdot \frac{0.5}{v} \]

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
 (*
  (pow
   (pow
    (pow
     (exp (+ 0.6931 (/ (fma cosTheta_i cosTheta_O -1.0) v)))
     0.16666666666666666)
    2.0)
   3.0)
  (/ 0.5 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 powf(powf(powf(expf((0.6931f + (fmaf(cosTheta_i, cosTheta_O, -1.0f) / v))), 0.16666666666666666f), 2.0f), 3.0f) * (0.5f / 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((((exp(Float32(Float32(0.6931) + Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) / v))) ^ Float32(0.16666666666666666)) ^ Float32(2.0)) ^ Float32(3.0)) * Float32(Float32(0.5) / 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

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

   [Start] 0.1	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   remove-double-neg [<=] 0.1	\[ e^{\color{blue}{\left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [<=] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   log-rec [=>] 0.1	\[ e^{\color{blue}{\left(-\log \left(2 \cdot v\right)\right)} + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} \]
   distribute-neg-in [<=] 0.1	\[ e^{\color{blue}{-\left(\log \left(2 \cdot v\right) + \left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   sub-neg [<=] 0.1	\[ e^{-\color{blue}{\left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   sub0-neg [<=] 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	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

3. Taylor expanded in sinTheta_i around 0 0.1

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

   \[\leadsto {\color{blue}{\left({\left({\left(e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)}^{0.16666666666666666}\right)}^{2}\right)}}^{3} \cdot \frac{0.5}{v} \]

6. Final simplification 0.1

   \[\leadsto {\left({\left({\left(e^{0.6931 + \frac{\mathsf{fma}\left(cosTheta_i, cosTheta_O, -1\right)}{v}}\right)}^{0.16666666666666666}\right)}^{2}\right)}^{3} \cdot \frac{0.5}{v} \]

Alternatives

Alternative 1
Error	0.2
Cost	9952

\[\frac{0.5}{v} \cdot {\left({\left(e^{0.6931 + \frac{-1}{v}}\right)}^{0.3333333333333333}\right)}^{3} \]

Alternative 2
Error	0.1
Cost	6688

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

Alternative 3
Error	0.1
Cost	3488

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

Alternative 4
Error	0.7
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	25.7
Cost	160

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

Alternative 7
Error	19.8
Cost	160

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

Alternative 8
Error	29.9
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023018 
(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))))))