HairBSDF, Mp, lower

Average Accuracy: 99.7% → 99.6%

Time: 18.3s

Precision: binary32

Cost: 9888

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

Math

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

↓

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

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
 (pow E (+ (+ 0.6931 (log (/ 0.5 v))) (/ -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 powf(((float) M_E), ((0.6931f + logf((0.5f / v))) + (-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 Float32(exp(1)) ^ Float32(Float32(Float32(0.6931) + log(Float32(Float32(0.5) / v))) + Float32(Float32(-1.0) / 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(2.71828182845904523536) ^ ((single(0.6931) + log((single(0.5) / v))) + (single(-1.0) / v));
end

Derivation

1. Initial program 99.7%

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

2. Simplified 99.7%

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

   [Start] 99.7	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   associate-+l+ [=>] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]
   sub-neg [=>] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [=>] 99.7	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [<=] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   sub-neg [<=] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate--l- [=>] 99.7	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/l* [=>] 99.7	\[ e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/r* [=>] 99.7	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]
   metadata-eval [=>] 99.7	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Applied egg-rr 99.7%

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

   [Start] 99.7	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \]
   *-un-lft-identity [=>] 99.7	\[ e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
   exp-prod [=>] 99.7	\[ \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
   exp-1-e [=>] 99.7	\[ {\color{blue}{e}}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]
   div-inv [=>] 99.7	\[ {e}^{\left(\left(\color{blue}{cosTheta_i \cdot \frac{1}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]
   clear-num [<=] 99.7	\[ {e}^{\left(\left(cosTheta_i \cdot \color{blue}{\frac{cosTheta_O}{v}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]
   +-commutative [=>] 99.7	\[ {e}^{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \color{blue}{\left(\frac{1}{v} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]
   *-un-lft-identity [=>] 99.7	\[ {e}^{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \left(\color{blue}{1 \cdot \frac{1}{v}} + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]
   div-inv [=>] 99.6	\[ {e}^{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \left(1 \cdot \frac{1}{v} + \color{blue}{\left(sinTheta_i \cdot sinTheta_O\right) \cdot \frac{1}{v}}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]
   distribute-rgt-out [=>] 99.7	\[ {e}^{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - \color{blue}{\frac{1}{v} \cdot \left(1 + sinTheta_i \cdot sinTheta_O\right)}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

4. Taylor expanded in sinTheta_i around 0 99.7%

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

5. Simplified 99.7%

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

   [Start] 99.7	\[ {e}^{\left(\left(0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}\right)} \]
   associate-+r+ [=>] 99.7	\[ {e}^{\left(\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)} - \frac{1}{v}\right)} \]
   associate-*r/ [<=] 99.7	\[ {e}^{\left(\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \color{blue}{cosTheta_i \cdot \frac{cosTheta_O}{v}}\right) - \frac{1}{v}\right)} \]

6. Taylor expanded in v around inf 99.6%

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

7. Simplified 99.6%

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

   [Start] 99.6	\[ {e}^{\left(\left(\log \left(\frac{1}{v}\right) + \left(0.6931 + \log 0.5\right)\right) - \frac{1}{v}\right)} \]
   +-commutative [=>] 99.6	\[ {e}^{\left(\color{blue}{\left(\left(0.6931 + \log 0.5\right) + \log \left(\frac{1}{v}\right)\right)} - \frac{1}{v}\right)} \]
   associate-+l+ [=>] 99.6	\[ {e}^{\left(\color{blue}{\left(0.6931 + \left(\log 0.5 + \log \left(\frac{1}{v}\right)\right)\right)} - \frac{1}{v}\right)} \]
   log-rec [=>] 99.7	\[ {e}^{\left(\left(0.6931 + \left(\log 0.5 + \color{blue}{\left(-\log v\right)}\right)\right) - \frac{1}{v}\right)} \]
   sub-neg [<=] 99.7	\[ {e}^{\left(\left(0.6931 + \color{blue}{\left(\log 0.5 - \log v\right)}\right) - \frac{1}{v}\right)} \]
   log-div [<=] 99.6	\[ {e}^{\left(\left(0.6931 + \color{blue}{\log \left(\frac{0.5}{v}\right)}\right) - \frac{1}{v}\right)} \]

8. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	6688

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

Alternative 2
Accuracy	19.7%
Cost	3524

\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -2.942726775082116 \cdot 10^{-44}:\\ \;\;\;\;e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta_O}{\frac{v}{sinTheta_i}}}\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	3488

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

Alternative 4
Accuracy	98.1%
Cost	3424

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

Alternative 5
Accuracy	12.5%
Cost	3360

\[e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}} \]

Alternative 6
Accuracy	6.4%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023144 
(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))))))