HairBSDF, Mp, lower

Average Error: 0.38% → 0.31%

Time: 20.2s

Precision: binary32

Cost: 16576

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{e^{0.6931 + \frac{-1}{v}}}\\ 0.5 \cdot \frac{\frac{{t_0}^{2}}{\frac{1}{t_0}}}{v} \end{array} \]

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
 (let* ((t_0 (cbrt (exp (+ 0.6931 (/ -1.0 v))))))
   (* 0.5 (/ (/ (pow t_0 2.0) (/ 1.0 t_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) {
	float t_0 = cbrtf(expf((0.6931f + (-1.0f / v))));
	return 0.5f * ((powf(t_0, 2.0f) / (1.0f / t_0)) / 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)
	t_0 = cbrt(exp(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))))
	return Float32(Float32(0.5) * Float32(Float32((t_0 ^ Float32(2.0)) / Float32(Float32(1.0) / t_0)) / v))
end

Derivation

1. Initial program 0.38

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

2. Simplified 0.4

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

   [Start] 0.38	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   remove-double-neg [<=] 0.38	\[ e^{\color{blue}{\left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [<=] 0.38	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   log-rec [=>] 0.27	\[ e^{\color{blue}{\left(-\log \left(2 \cdot v\right)\right)} + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} \]
   distribute-neg-in [<=] 0.27	\[ e^{\color{blue}{-\left(\log \left(2 \cdot v\right) + \left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   sub-neg [<=] 0.27	\[ e^{-\color{blue}{\left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   sub0-neg [<=] 0.27	\[ e^{\color{blue}{0 - \left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   associate-+l- [<=] 0.27	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

3. Taylor expanded in cosTheta_i around 0 0.39

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

4. Simplified 0.39

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

   [Start] 0.39	\[ e^{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   associate-*l/ [<=] 0.39	\[ e^{-1 \cdot \color{blue}{\left(\frac{sinTheta_i}{v} \cdot sinTheta_O\right)} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   mul-1-neg [=>] 0.39	\[ e^{\color{blue}{\left(-\frac{sinTheta_i}{v} \cdot sinTheta_O\right)} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   distribute-rgt-neg-out [<=] 0.39	\[ e^{\color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]

5. Taylor expanded in sinTheta_i around 0 0.31

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

6. Applied egg-rr 0.31

   \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{e^{0.6931 + \frac{-1}{v}}}\right)}^{2}}{\frac{1}{\sqrt[3]{e^{0.6931 + \frac{-1}{v}}}}}}}{v} \]

7. Final simplification 0.31

   \[\leadsto 0.5 \cdot \frac{\frac{{\left(\sqrt[3]{e^{0.6931 + \frac{-1}{v}}}\right)}^{2}}{\frac{1}{\sqrt[3]{e^{0.6931 + \frac{-1}{v}}}}}}{v} \]

Alternatives

Alternative 1
Error	0.3%
Cost	9920

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

Alternative 2
Error	0.31%
Cost	3488

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

Alternative 3
Error	2%
Cost	3424

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

Alternative 4
Error	2.22%
Cost	3360

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

Alternative 5
Error	90.29%
Cost	288

\[-0.5 \cdot \left(sinTheta_O \cdot \frac{sinTheta_i}{v \cdot v}\right) \]

Alternative 6
Error	86.25%
Cost	288

\[\frac{-0.5}{v} \cdot \left(sinTheta_O \cdot \frac{sinTheta_i}{v}\right) \]

Alternative 7
Error	95.33%
Cost	96

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

Reproduce

herbie shell --seed 2023089 
(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))))))