HairBSDF, Mp, upper

Average Accuracy: 98.6% → 98.8%

Time: 23.1s

Precision: binary32

Cost: 7136

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

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

Math

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

↓

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

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
 (*
  (*
   (/ 1.0 (sinh (/ 1.0 v)))
   (* (/ 0.5 v) (/ cosTheta_O (exp (/ (* sinTheta_i sinTheta_O) v)))))
  (* (/ 1.0 v) cosTheta_i)))

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 ((1.0f / sinhf((1.0f / v))) * ((0.5f / v) * (cosTheta_O / expf(((sinTheta_i * sinTheta_O) / v))))) * ((1.0f / v) * cosTheta_i);
}

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 = ((1.0e0 / sinh((1.0e0 / v))) * ((0.5e0 / v) * (costheta_o / exp(((sintheta_i * sintheta_o) / v))))) * ((1.0e0 / v) * costheta_i)
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(Float32(1.0) / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(Float32(0.5) / v) * Float32(cosTheta_O / exp(Float32(Float32(sinTheta_i * sinTheta_O) / v))))) * Float32(Float32(Float32(1.0) / v) * cosTheta_i))
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 = ((single(1.0) / sinh((single(1.0) / v))) * ((single(0.5) / v) * (cosTheta_O / exp(((sinTheta_i * sinTheta_O) / v))))) * ((single(1.0) / v) * cosTheta_i);
end

Derivation

1. Initial program 98.6%

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

2. Simplified 98.6%

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

   [Start] 98.6	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.6	\[ \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*r/ [<=] 98.6	\[ \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]
   associate-/l* [=>] 98.7	\[ \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-/r/ [=>] 98.6	\[ \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.6	\[ \color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   *-commutative [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]
   associate-*r* [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]
   associate-/l/ [<=] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]
   exp-neg [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-/l/ [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-/r* [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   metadata-eval [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   associate-*l/ [<=] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   *-commutative [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   exp-prod [=>] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Applied egg-rr 98.4%

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

   [Start] 98.6	\[ \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]
   *-commutative [=>] 98.6	\[ \color{blue}{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right)} \]
   associate-*r/ [=>] 98.6	\[ \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \color{blue}{\frac{cosTheta_O \cdot cosTheta_i}{v}} \]
   associate-*r/ [=>] 98.2	\[ \color{blue}{\frac{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{v}} \]
   associate-/r* [=>] 98.4	\[ \frac{\color{blue}{\frac{\frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v}}{\sinh \left(\frac{1}{v}\right)}} \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{v} \]
   associate-/l/ [=>] 98.4	\[ \frac{\frac{\color{blue}{\frac{0.5}{v \cdot {\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{v} \]
   pow-exp [=>] 98.4	\[ \frac{\frac{\frac{0.5}{v \cdot \color{blue}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{v} \]

4. Applied egg-rr 99.0%

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

   [Start] 98.4	\[ \frac{\frac{\frac{0.5}{v \cdot e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{v} \]
   div-inv [=>] 98.4	\[ \color{blue}{\left(\frac{\frac{0.5}{v \cdot e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_O \cdot cosTheta_i\right)\right) \cdot \frac{1}{v}} \]
   associate-*r* [=>] 98.5	\[ \color{blue}{\left(\left(\frac{\frac{0.5}{v \cdot e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right) \cdot cosTheta_i\right)} \cdot \frac{1}{v} \]
   associate-*l* [=>] 99.0	\[ \color{blue}{\left(\frac{\frac{0.5}{v \cdot e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right)} \]
   associate-/r* [=>] 99.0	\[ \left(\frac{\color{blue}{\frac{\frac{0.5}{v}}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   exp-prod [=>] 99.0	\[ \left(\frac{\frac{\frac{0.5}{v}}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]

5. Applied egg-rr 60.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.5}{e^{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \log v\right)}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)} - 1\right)} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   Proof

   [Start] 99.0	\[ \left(\frac{\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   expm1-log1p-u [=>] 99.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right)\right)} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   expm1-udef [=>] 60.3	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right)} - 1\right)} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-*l/ [=>] 60.2	\[ \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}} \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right)}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-/l* [=>] 60.3	\[ \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\frac{0.5}{v}}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-/l/ [=>] 60.3	\[ \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)} \cdot v}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   pow-exp [=>] 60.3	\[ \left(e^{\mathsf{log1p}\left(\frac{\frac{0.5}{\color{blue}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}}} \cdot v}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   add-exp-log [=>] 60.3	\[ \left(e^{\mathsf{log1p}\left(\frac{\frac{0.5}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v}} \cdot \color{blue}{e^{\log v}}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   prod-exp [=>] 60.3	\[ \left(e^{\mathsf{log1p}\left(\frac{\frac{0.5}{\color{blue}{e^{sinTheta_O \cdot \frac{sinTheta_i}{v} + \log v}}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   fma-def [=>] 60.3	\[ \left(e^{\mathsf{log1p}\left(\frac{\frac{0.5}{e^{\color{blue}{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \log v\right)}}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]

6. Simplified 98.9%

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

   [Start] 60.3	\[ \left(e^{\mathsf{log1p}\left(\frac{\frac{0.5}{e^{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \log v\right)}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)} - 1\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   expm1-def [=>] 98.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.5}{e^{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \log v\right)}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}\right)\right)} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   expm1-log1p [=>] 98.6	\[ \color{blue}{\frac{\frac{0.5}{e^{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \log v\right)}}}{\frac{\sinh \left(\frac{1}{v}\right)}{cosTheta_O}}} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-/r/ [=>] 99.0	\[ \color{blue}{\left(\frac{\frac{0.5}{e^{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \log v\right)}}}{\sinh \left(\frac{1}{v}\right)} \cdot cosTheta_O\right)} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-*l/ [=>] 98.9	\[ \color{blue}{\frac{\frac{0.5}{e^{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \log v\right)}} \cdot cosTheta_O}{\sinh \left(\frac{1}{v}\right)}} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]

7. Applied egg-rr 99.0%

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

   [Start] 98.9	\[ \frac{\frac{0.5}{v} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   clear-num [=>] 98.7	\[ \color{blue}{\frac{1}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}}}} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-/r/ [=>] 99.0	\[ \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_i}{\frac{v}{sinTheta_O}}}}\right)\right)} \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-/r/ [=>] 99.0	\[ \left(\frac{1}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta_O}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}\right)\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]
   associate-*l/ [=>] 99.0	\[ \left(\frac{1}{\sinh \left(\frac{1}{v}\right)} \cdot \left(\frac{0.5}{v} \cdot \frac{cosTheta_O}{e^{\color{blue}{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}\right)\right) \cdot \left(cosTheta_i \cdot \frac{1}{v}\right) \]

8. Final simplification 99.0%

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

Alternatives

Alternative 1
Accuracy	98.9%
Cost	7072

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

Alternative 2
Accuracy	98.7%
Cost	7008

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

Alternative 3
Accuracy	98.6%
Cost	3680

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

Alternative 4
Accuracy	98.6%
Cost	3680

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

Alternative 5
Accuracy	98.7%
Cost	3680

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

Alternative 6
Accuracy	98.2%
Cost	3616

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

Alternative 7
Accuracy	98.5%
Cost	3616

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

Alternative 8
Accuracy	98.6%
Cost	3616

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

Alternative 9
Accuracy	98.5%
Cost	3616

\[\frac{\frac{cosTheta_O}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}}{\frac{v}{cosTheta_i}} \]

Alternative 10
Accuracy	64.0%
Cost	992

\[\frac{cosTheta_i \cdot \frac{cosTheta_O}{v}}{2 + 2 \cdot \left(sinTheta_O \cdot \frac{sinTheta_i}{v} + \frac{0.16666666666666666 + 0.5 \cdot \left(\left(sinTheta_i \cdot sinTheta_i\right) \cdot \left(sinTheta_O \cdot sinTheta_O\right)\right)}{v \cdot v}\right)} \]

Alternative 11
Accuracy	58.8%
Cost	288

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

Alternative 12
Accuracy	58.2%
Cost	224

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

Alternative 13
Accuracy	58.2%
Cost	224

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

Alternative 14
Accuracy	58.2%
Cost	224

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

Alternative 15
Accuracy	58.2%
Cost	224

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

Alternative 16
Accuracy	58.8%
Cost	224

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

Reproduce

herbie shell --seed 2023157 
(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)))