HairBSDF, Mp, lower

Average Accuracy: 99.6% → 99.5%

Time: 17.0s

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

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

↓

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

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 E (/ -1.0 v)) (exp 0.6931)) (/ 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(((float) M_E), (-1.0f / v)) * expf(0.6931f)) * (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(Float32((Float32(exp(1)) ^ Float32(Float32(-1.0) / v)) * exp(Float32(0.6931))) * Float32(Float32(0.5) / 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(-1.0) / v)) * exp(single(0.6931))) * (single(0.5) / v);
end

Derivation

1. Initial program 99.6%

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

2. Simplified 99.6%

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

   [Start] 99.6	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   remove-double-neg [<=] 99.6	\[ e^{\color{blue}{\left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [<=] 99.6	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   log-rec [=>] 99.8	\[ e^{\color{blue}{\left(-\log \left(2 \cdot v\right)\right)} + \left(-\left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)} \]
   distribute-neg-in [<=] 99.8	\[ e^{\color{blue}{-\left(\log \left(2 \cdot v\right) + \left(-\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)\right)}} \]
   sub-neg [<=] 99.8	\[ e^{-\color{blue}{\left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   sub0-neg [<=] 99.8	\[ e^{\color{blue}{0 - \left(\log \left(2 \cdot v\right) - \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)\right)}} \]
   associate-+l- [<=] 99.8	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

3. Applied egg-rr 99.6%

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

4. Applied egg-rr 99.6%

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

5. Simplified 99.6%

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

   [Start] 99.6	\[ \left({\left(e^{1}\right)}^{\left(\frac{1}{v} \cdot \left(\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1\right)\right)} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v} \]
   exp-1-e [=>] 99.6	\[ \left({\color{blue}{e}}^{\left(\frac{1}{v} \cdot \left(\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1\right)\right)} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v} \]
   +-commutative [=>] 99.6	\[ \left({e}^{\left(\frac{1}{v} \cdot \color{blue}{\left(-1 + \left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right)\right)}\right)} \cdot e^{0.6931}\right) \cdot \frac{0.5}{v} \]

6. Taylor expanded in sinTheta_i around 0 99.6%

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

7. Taylor expanded in cosTheta_i around 0 99.5%

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

8. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6688

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

Alternative 2
Accuracy	52.5%
Cost	3656

\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 6.000000142453368 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta_i\right) \cdot \frac{sinTheta_O}{v}}\\ \end{array} \]

Alternative 3
Accuracy	52.5%
Cost	3656

\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 6.000000142453368 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}\\ \end{array} \]

Alternative 4
Accuracy	52.5%
Cost	3656

\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 6.000000142453368 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}\\ \end{array} \]

Alternative 5
Accuracy	45.7%
Cost	3492

\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \end{array} \]

Alternative 6
Accuracy	45.7%
Cost	3492

\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \end{array} \]

Alternative 7
Accuracy	99.6%
Cost	3488

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

Alternative 8
Accuracy	99.7%
Cost	3488

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

Alternative 9
Accuracy	39.2%
Cost	224

\[\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}} \]

Alternative 10
Accuracy	19.8%
Cost	160

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

Alternative 11
Accuracy	38.9%
Cost	160

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

Alternative 12
Accuracy	6.4%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023125 
(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))))))