HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 15.0s

Precision: binary32

Cost: 3744

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

Math

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

↓

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

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
 (*
  (* (+ 1.0 (/ (* cosTheta_i cosTheta_O) v)) (exp (+ (/ -1.0 v) 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 ((1.0f + ((cosTheta_i * cosTheta_O) / v)) * expf(((-1.0f / v) + 0.6931f))) * (0.5f / 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 = ((1.0e0 + ((costheta_i * costheta_o) / v)) * exp((((-1.0e0) / v) + 0.6931e0))) * (0.5e0 / 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 Float32(Float32(Float32(Float32(1.0) + Float32(Float32(cosTheta_i * cosTheta_O) / v)) * exp(Float32(Float32(Float32(-1.0) / v) + 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(1.0) + ((cosTheta_i * cosTheta_O) / v)) * exp(((single(-1.0) / v) + single(0.6931)))) * (single(0.5) / v);
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	\[ e^{\color{blue}{\left(0 - \log \left(2 \cdot v\right)\right) + \left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)}} \]

3. Taylor expanded in cosTheta_i around 0 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ e^{\left(-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   +-commutative [=>] 0.1	\[ e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} + -1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}\right)} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   associate-*l/ [<=] 0.1	\[ e^{\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} + -1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   associate-*l/ [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O + -1 \cdot \color{blue}{\left(\frac{sinTheta_i}{v} \cdot sinTheta_O\right)}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   mul-1-neg [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O + \color{blue}{\left(-\frac{sinTheta_i}{v} \cdot sinTheta_O\right)}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   distribute-rgt-neg-out [<=] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O + \color{blue}{\frac{sinTheta_i}{v} \cdot \left(-sinTheta_O\right)}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   distribute-rgt-neg-out [=>] 0.1	\[ e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O + \color{blue}{\left(-\frac{sinTheta_i}{v} \cdot sinTheta_O\right)}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   sub-neg [<=] 0.1	\[ e^{\color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right)} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   associate-*l/ [=>] 0.1	\[ e^{\left(\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v}} - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   associate-*l/ [=>] 0.1	\[ e^{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]
   div-sub [<=] 0.1	\[ e^{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}} + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v} \]

5. Applied egg-rr 4.3

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

6. Taylor expanded in v around inf 0.1

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

7. Simplified 0.1

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

   [Start] 0.1	\[ \left(\left(\left(1 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right) - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) \cdot e^{\frac{-1}{v} + 0.6931}\right) \cdot \frac{0.5}{v} \]
   associate--l+ [=>] 0.1	\[ \left(\color{blue}{\left(1 + \left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right)} \cdot e^{\frac{-1}{v} + 0.6931}\right) \cdot \frac{0.5}{v} \]
   div-sub [<=] 0.1	\[ \left(\left(1 + \color{blue}{\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v}}\right) \cdot e^{\frac{-1}{v} + 0.6931}\right) \cdot \frac{0.5}{v} \]

8. Taylor expanded in cosTheta_i around inf 0.1

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

9. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	3616

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

Alternative 2
Error	0.1
Cost	3488

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

Alternative 3
Error	0.1
Cost	3488

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

Alternative 4
Error	0.6
Cost	3424

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

Alternative 5
Error	30.5
Cost	3360

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

Reproduce

herbie shell --seed 2023034 
(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))))))