HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 16.1s

Precision: binary32

Cost: 16512

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

Math

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

↓

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

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 (- v))))
   (/ (/ (* -0.5 (pow E (+ 0.6931 (/ -1.0 v)))) (* t_0 t_0)) t_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) {
	float t_0 = cbrtf(-v);
	return ((-0.5f * powf(((float) M_E), (0.6931f + (-1.0f / v)))) / (t_0 * t_0)) / t_0;
}

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(Float32(-v))
	return Float32(Float32(Float32(Float32(-0.5) * (Float32(exp(1)) ^ Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))) / Float32(t_0 * t_0)) / t_0)
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	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   metadata-eval [<=] 0.1	\[ e^{\left(\color{blue}{\log 1} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   log-div [<=] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   exp-sum [=>] 0.1	\[ \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]
   *-commutative [<=] 0.1	\[ \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]
   *-rgt-identity [<=] 0.1	\[ \color{blue}{\left(e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot 1\right)} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)} \]
   associate-*r* [<=] 0.1	\[ \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right)} \]
   +-lft-identity [<=] 0.1	\[ e^{\color{blue}{0 + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   associate-+l- [=>] 0.1	\[ e^{0 + \color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \left(\frac{1}{v} - 0.6931\right)\right)}} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   sub-neg [=>] 0.1	\[ e^{0 + \color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\left(\frac{1}{v} - 0.6931\right)\right)\right)}} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   associate-+r+ [=>] 0.1	\[ e^{\color{blue}{\left(0 + \left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right) + \left(-\left(\frac{1}{v} - 0.6931\right)\right)}} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   +-lft-identity [=>] 0.1	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right)} + \left(-\left(\frac{1}{v} - 0.6931\right)\right)} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   associate-*l/ [<=] 0.1	\[ e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}\right) + \left(-\left(\frac{1}{v} - 0.6931\right)\right)} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   associate-*l/ [<=] 0.1	\[ e^{\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(-\left(\frac{1}{v} - 0.6931\right)\right)} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   neg-sub0 [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \color{blue}{\left(0 - \left(\frac{1}{v} - 0.6931\right)\right)}} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   associate-+l- [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \color{blue}{\left(\left(0 - \frac{1}{v}\right) + 0.6931\right)}} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   neg-sub0 [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\color{blue}{\left(-\frac{1}{v}\right)} + 0.6931\right)} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   distribute-neg-frac [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\color{blue}{\frac{-1}{v}} + 0.6931\right)} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   metadata-eval [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{\color{blue}{-1}}{v} + 0.6931\right)} \cdot \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right) \]
   *-lft-identity [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]
   rem-exp-log [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]
   associate-/r* [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]
   metadata-eval [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{\color{blue}{0.5}}{v} \]

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. Taylor expanded in cosTheta_i around 0 0.1

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

5. Applied egg-rr 0.1

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

6. Applied egg-rr 0.1

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

7. Simplified 0.1

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

   [Start] 0.1	\[ \frac{\frac{\left({\left(e^{1}\right)}^{\left(\frac{0.6931 + \frac{-1}{v}}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{0.6931 + \frac{-1}{v}}{2}\right)}\right) \cdot -0.5}{\sqrt[3]{-v} \cdot \sqrt[3]{-v}}}{\sqrt[3]{-v}} \]
   sqr-pow [<=] 0.1	\[ \frac{\frac{\color{blue}{{\left(e^{1}\right)}^{\left(0.6931 + \frac{-1}{v}\right)}} \cdot -0.5}{\sqrt[3]{-v} \cdot \sqrt[3]{-v}}}{\sqrt[3]{-v}} \]
   exp-1-e [=>] 0.1	\[ \frac{\frac{{\color{blue}{e}}^{\left(0.6931 + \frac{-1}{v}\right)} \cdot -0.5}{\sqrt[3]{-v} \cdot \sqrt[3]{-v}}}{\sqrt[3]{-v}} \]

8. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	6688

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

Alternative 2
Error	0.1
Cost	3488

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

Alternative 3
Error	0.6
Cost	3424

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

Alternative 4
Error	0.6
Cost	3296

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

Alternative 5
Error	19.5
Cost	192

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

Alternative 6
Error	25.5
Cost	160

\[sinTheta_O \cdot \frac{sinTheta_i}{v} \]

Alternative 7
Error	25.5
Cost	160

\[sinTheta_i \cdot \frac{sinTheta_O}{v} \]

Alternative 8
Error	19.5
Cost	160

\[\frac{sinTheta_O \cdot sinTheta_i}{v} \]

Alternative 9
Error	29.9
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022354 
(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))))))