HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 16.3s

Precision: binary32

Cost: 7008

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

\[ \begin{array}{c}[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \end{array} \]

Math

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

↓

\[\left(\frac{cosTheta_i}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* (/ cosTheta_i v) (/ (/ 0.5 v) (sinh (/ 1.0 v))))
  (/ cosTheta_O (exp (/ sinTheta_i (/ v sinTheta_O))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i / v) * ((0.5f / v) / sinhf((1.0f / v)))) * (cosTheta_O / expf((sinTheta_i / (v / sinTheta_O))));
}

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(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 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 = ((costheta_i / v) * ((0.5e0 / v) / sinh((1.0e0 / v)))) * (costheta_o / exp((sintheta_i / (v / sintheta_o))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

↓

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i / v) * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v)))) * Float32(cosTheta_O / exp(Float32(sinTheta_i / Float32(v / sinTheta_O)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

↓

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / v) * ((single(0.5) / v) / sinh((single(1.0) / v)))) * (cosTheta_O / exp((sinTheta_i / (v / sinTheta_O))));
end

Derivation

1. Initial program 0.5

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

2. Simplified 0.5

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]
   Proof

   [Start] 0.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-/l* [=>] 0.5	\[ \color{blue}{\frac{\frac{cosTheta_i \cdot cosTheta_O}{v}}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}} \]
   associate-/l/ [=>] 0.5	\[ \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\frac{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}} \cdot v}} \]
   associate-*l/ [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}} \]
   exp-neg [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}} \]
   associate-/r/ [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}{1} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]
   /-rgt-identity [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   associate-*l* [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   associate-*l* [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]
   associate-*l/ [<=] 0.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

3. Applied egg-rr 15.3

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right)}}{v \cdot \left(v \cdot 2\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}\right)} - 1} \]

4. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta_i}{v}}{v \cdot 2}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}} \]
   Proof

   [Start] 15.3	\[ e^{\mathsf{log1p}\left(\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right)}}{v \cdot \left(v \cdot 2\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}\right)} - 1 \]
   expm1-def [=>] 0.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right)}}{v \cdot \left(v \cdot 2\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}\right)\right)} \]
   expm1-log1p [=>] 0.6	\[ \color{blue}{\frac{\frac{\frac{cosTheta_i \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right)}}{v \cdot \left(v \cdot 2\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}} \]
   associate-/l/ [=>] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot \sinh \left(\frac{1}{v}\right)}}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \]
   *-lft-identity [<=] 0.5	\[ \frac{\frac{\color{blue}{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}}{\left(v \cdot \left(v \cdot 2\right)\right) \cdot \sinh \left(\frac{1}{v}\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \]
   *-commutative [=>] 0.5	\[ \frac{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(v \cdot \color{blue}{\left(2 \cdot v\right)}\right) \cdot \sinh \left(\frac{1}{v}\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \]
   associate-*r* [<=] 0.5	\[ \frac{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\color{blue}{v \cdot \left(\left(2 \cdot v\right) \cdot \sinh \left(\frac{1}{v}\right)\right)}}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \]
   *-commutative [<=] 0.5	\[ \frac{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot \color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)}}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \]
   *-commutative [<=] 0.5	\[ \frac{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot 2\right)}\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \]
   associate-*l/ [<=] 0.5	\[ \frac{\color{blue}{\frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}} \]
   associate-*r/ [<=] 0.5	\[ \color{blue}{\frac{1}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \cdot \frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}} \]
   associate-*l/ [=>] 0.5	\[ \color{blue}{\frac{1 \cdot \frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)}} \]
   *-lft-identity [=>] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{{\left(e^{sinTheta_i}\right)}^{\left(\frac{sinTheta_O}{v}\right)}}}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)} \]

5. Applied egg-rr 12.4

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \left(v \cdot v\right)\right)}\right)} - 1\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]

6. Simplified 0.4

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

   [Start] 12.4	\[ \left(e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \left(v \cdot v\right)\right)}\right)} - 1\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   expm1-def [=>] 0.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   expm1-log1p [=>] 0.5	\[ \color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   *-rgt-identity [<=] 0.5	\[ \frac{\color{blue}{cosTheta_i \cdot 1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \left(v \cdot v\right)\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   times-frac [=>] 0.5	\[ \color{blue}{\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{1}{2 \cdot \left(v \cdot v\right)}\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-*r* [=>] 0.5	\[ \left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{1}{\color{blue}{\left(2 \cdot v\right) \cdot v}}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   *-commutative [<=] 0.5	\[ \left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{1}{\color{blue}{\left(v \cdot 2\right)} \cdot v}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-/l/ [<=] 0.4	\[ \left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{\frac{1}{v}}{v \cdot 2}}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-*r/ [=>] 0.4	\[ \color{blue}{\frac{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{1}{v}}{v \cdot 2}} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-*l/ [=>] 0.4	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot \frac{1}{v}}{\sinh \left(\frac{1}{v}\right)}}}{v \cdot 2} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-/r* [<=] 0.4	\[ \color{blue}{\frac{cosTheta_i \cdot \frac{1}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-*r/ [=>] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot 1}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-*l/ [<=] 0.5	\[ \frac{\color{blue}{\frac{cosTheta_i}{v} \cdot 1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   *-commutative [<=] 0.5	\[ \frac{\frac{cosTheta_i}{v} \cdot 1}{\color{blue}{\left(v \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)}} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-*r/ [<=] 0.5	\[ \color{blue}{\left(\frac{cosTheta_i}{v} \cdot \frac{1}{\left(v \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)}\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-/r* [=>] 0.4	\[ \left(\frac{cosTheta_i}{v} \cdot \color{blue}{\frac{\frac{1}{v \cdot 2}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   *-commutative [=>] 0.4	\[ \left(\frac{cosTheta_i}{v} \cdot \frac{\frac{1}{\color{blue}{2 \cdot v}}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   associate-/r* [=>] 0.4	\[ \left(\frac{cosTheta_i}{v} \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]
   metadata-eval [=>] 0.4	\[ \left(\frac{cosTheta_i}{v} \cdot \frac{\frac{\color{blue}{0.5}}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]

7. Final simplification 0.4

   \[\leadsto \left(\frac{cosTheta_i}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \]

Alternatives

Alternative 1
Error	0.4
Cost	7008

\[\frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}} \cdot \left(cosTheta_i \cdot \frac{\frac{\frac{0.5}{v}}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

Alternative 2
Error	0.6
Cost	3616

\[\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

Alternative 3
Error	0.5
Cost	3616

\[\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \]

Alternative 4
Error	13.3
Cost	416

\[\frac{1}{v} \cdot \frac{1}{\frac{\frac{1}{cosTheta_O}}{cosTheta_i \cdot 0.5}} \]

Alternative 5
Error	11.7
Cost	416

\[\frac{cosTheta_i}{2 \cdot \frac{v}{cosTheta_O} + \frac{0.3333333333333333}{v \cdot cosTheta_O}} \]

Alternative 6
Error	13.5
Cost	224

\[0.5 \cdot \left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right) \]

Alternative 7
Error	13.5
Cost	224

\[0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]

Alternative 8
Error	13.5
Cost	224

\[0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}} \]

Alternative 9
Error	13.3
Cost	224

\[\frac{0.5}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]

Reproduce

herbie shell --seed 2023066 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (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))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))