HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 15.8s

Precision: binary32

Cost: 22880

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

Math

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

↓

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

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
  (sqrt
   (pow
    (cbrt
     (exp
      (+
       0.6931
       (+ (fma cosTheta_i (/ cosTheta_O v) (log (/ 0.5 v))) (/ -1.0 v)))))
    3.0))
  2.0))

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) {
	return powf(sqrtf(powf(cbrtf(expf((0.6931f + (fmaf(cosTheta_i, (cosTheta_O / v), logf((0.5f / v))) + (-1.0f / v))))), 3.0f)), 2.0f);
}

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 sqrt((cbrt(exp(Float32(Float32(0.6931) + Float32(fma(cosTheta_i, Float32(cosTheta_O / v), log(Float32(Float32(0.5) / v))) + Float32(Float32(-1.0) / v))))) ^ Float32(3.0))) ^ Float32(2.0)
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

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

   [Start] 0.1	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [=>] 0.1	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   log-div [=>] 0.1	\[ e^{\color{blue}{\left(\log 1 - \log \left(2 \cdot v\right)\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   metadata-eval [=>] 0.1	\[ e^{\left(\color{blue}{0} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   associate-+l- [=>] 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	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   +-commutative [<=] 0.1	\[ e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]
   associate-+l+ [=>] 0.1	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]
   sub-neg [=>] 0.1	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   sub-neg [=>] 0.1	\[ e^{\left(\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + \left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right)} + \left(-\frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l+ [=>] 0.1	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + \left(\left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l+ [<=] 0.1	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + \left(-\frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   sub-neg [<=] 0.1	\[ e^{\left(\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right)} + \left(-\frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-+l- [=>] 0.1	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   neg-mul-1 [=>] 0.1	\[ e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \color{blue}{-1 \cdot \frac{1}{v}}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   metadata-eval [<=] 0.1	\[ e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \color{blue}{\left(-1\right)} \cdot \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   cancel-sign-sub-inv [=>] 0.1	\[ e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \left(-\left(-1\right)\right) \cdot \frac{1}{v}\right)}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/l* [=>] 0.1	\[ e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \left(-\left(-1\right)\right) \cdot \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   associate-/l* [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\color{blue}{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}} + \left(-\left(-1\right)\right) \cdot \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   metadata-eval [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \left(-\color{blue}{-1}\right) \cdot \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   metadata-eval [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \color{blue}{1} \cdot \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   *-lft-identity [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \color{blue}{\frac{1}{v}}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]
   +-lft-identity [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \color{blue}{\left(0 + \log \left(\frac{1}{2 \cdot v}\right)\right)}\right)} \]
   metadata-eval [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \left(\color{blue}{\log 1} + \log \left(\frac{1}{2 \cdot v}\right)\right)\right)} \]
   rem-log-exp [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \left(\log 1 + \color{blue}{\log \left(e^{\log \left(\frac{1}{2 \cdot v}\right)}\right)}\right)\right)} \]
   log-prod [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \color{blue}{\log \left(1 \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}\right)}\right)} \]
   *-lft-identity [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(e^{\log \left(\frac{1}{2 \cdot v}\right)}\right)}\right)} \]
   rem-exp-log [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{1}{2 \cdot v}\right)}\right)} \]
   associate-/r* [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]
   metadata-eval [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i}{\frac{v}{sinTheta_O}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Taylor expanded in sinTheta_i around 0 0.1

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13120

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

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	0.6
Cost	3424

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

Alternative 5
Error	0.6
Cost	3296

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

Alternative 6
Error	19.5
Cost	224

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

Alternative 7
Error	19.5
Cost	192

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

Alternative 8
Error	25.5
Cost	160

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

Alternative 9
Error	25.5
Cost	160

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

Alternative 10
Error	19.5
Cost	160

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

Alternative 11
Error	30.0
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022356 
(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))))))