HairBSDF, Mp, lower

Average Error: 99.7% → 99.7%

Time: 16.5s

Precision: binary32

Cost: 6688.00

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

Math

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

↓

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

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
 (exp (+ 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 expf((0.6931f + (logf((0.5f / v)) + (-1.0f / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

↓

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((0.6931e0 + (log((0.5e0 / v)) + ((-1.0e0) / v))))
end function

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 exp(Float32(Float32(0.6931) + Float32(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 = exp((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

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

   [Start] 99.7	\[ e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
   +-commutative [=>] 99.7	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   log-div [=>] 99.8	\[ e^{\color{blue}{\left(\log 1 - \log \left(2 \cdot v\right)\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   metadata-eval [=>] 99.8	\[ e^{\left(\color{blue}{0} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   associate-+l- [=>] 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	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]
   metadata-eval [<=] 99.8	\[ e^{\left(\color{blue}{\log 1} - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   log-div [<=] 99.7	\[ e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)} + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]
   +-commutative [<=] 99.7	\[ e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]
   associate-+l+ [=>] 99.7	\[ e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

3. Taylor expanded in sinTheta_i around 0 99.7

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

4. Simplified 99.7

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

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

5. Taylor expanded in cosTheta_i around 0 99.7

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

6. Final simplification 99.7

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

Alternatives

Alternative 1
Error	99.7%
Cost	3488.00

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

Alternative 2
Error	98.2%
Cost	3296.00

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

Alternative 3
Error	62.4%
Cost	1036.00

\[\begin{array}{l} t_0 := \frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.401298464324817 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 0:\\ \;\;\;\;\frac{sinTheta_i \cdot sinTheta_O}{v}\\ \mathbf{elif}\;sinTheta_i \cdot sinTheta_O \leq 2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{\frac{-v}{sinTheta_i \cdot sinTheta_O} - \frac{v \cdot v}{sinTheta_O \cdot \left(sinTheta_O \cdot \left(sinTheta_i \cdot sinTheta_i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	58.5%
Cost	489.00

\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44} \lor \neg \left(cosTheta_i \cdot cosTheta_O \leq 1.401298464324817 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{1}{\frac{v}{sinTheta_i \cdot sinTheta_O}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]

Alternative 5
Error	57.8%
Cost	425.00

\[\begin{array}{l} \mathbf{if}\;sinTheta_i \cdot sinTheta_O \leq -1.401298464324817 \cdot 10^{-45} \lor \neg \left(sinTheta_i \cdot sinTheta_O \leq 5.002635517639597 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_i \cdot sinTheta_O}{v}\\ \end{array} \]

Alternative 6
Error	42.9%
Cost	292.00

\[\begin{array}{l} \mathbf{if}\;cosTheta_i \cdot cosTheta_O \leq -1.961817850054744 \cdot 10^{-44}:\\ \;\;\;\;\frac{sinTheta_O}{\frac{v}{sinTheta_i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{v}\\ \end{array} \]

Alternative 7
Error	22.5%
Cost	228.00

\[\begin{array}{l} \mathbf{if}\;cosTheta_i \leq -2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;sinTheta_i \cdot \frac{sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_i \cdot \frac{cosTheta_O}{v}\\ \end{array} \]

Alternative 8
Error	22.5%
Cost	228.00

\[\begin{array}{l} \mathbf{if}\;cosTheta_i \leq -2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;sinTheta_i \cdot \frac{sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \end{array} \]

Alternative 9
Error	22.5%
Cost	228.00

\[\begin{array}{l} \mathbf{if}\;cosTheta_i \leq -2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;sinTheta_i \cdot \frac{sinTheta_O}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v}{cosTheta_O}}\\ \end{array} \]

Alternative 10
Error	22.5%
Cost	228.00

\[\begin{array}{l} \mathbf{if}\;cosTheta_i \leq -2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{sinTheta_O}{\frac{v}{sinTheta_i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v}{cosTheta_O}}\\ \end{array} \]

Alternative 11
Error	20.1%
Cost	160.00

\[cosTheta_i \cdot \frac{cosTheta_O}{v} \]

Alternative 12
Error	6.4%
Cost	32.00

\[1 \]

Reproduce

herbie shell --seed 2023093 
(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))))))