HairBSDF, Mp, upper

Average Accuracy: 98.5% → 98.7%

Time: 19.0s

Precision: binary32

Cost: 7008

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

Math

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

↓

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

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 cosTheta_O)
  (*
   (/ (* v (sinh (/ 1.0 v))) (/ 0.5 v))
   (exp (* (/ sinTheta_O v) sinTheta_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 (cosTheta_i * cosTheta_O) / (((v * sinhf((1.0f / v))) / (0.5f / v)) * expf(((sinTheta_O / v) * sinTheta_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 = (costheta_i * costheta_o) / (((v * sinh((1.0e0 / v))) / (0.5e0 / v)) * exp(((sintheta_o / v) * sintheta_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(cosTheta_i * cosTheta_O) / Float32(Float32(Float32(v * sinh(Float32(Float32(1.0) / v))) / Float32(Float32(0.5) / v)) * exp(Float32(Float32(sinTheta_O / v) * sinTheta_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 = (cosTheta_i * cosTheta_O) / (((v * sinh((single(1.0) / v))) / (single(0.5) / v)) * exp(((sinTheta_O / v) * sinTheta_i)));
end

Derivation

1. Initial program 98.5%

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

2. Simplified 98.5%

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

   [Start] 98.5	\[ \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   associate-*l/ [<=] 98.5	\[ \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
   times-frac [<=] 98.5	\[ \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]
   exp-neg [=>] 98.5	\[ \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]
   associate-*l/ [=>] 98.5	\[ \frac{\color{blue}{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]
   *-lft-identity [=>] 98.5	\[ \frac{\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]
   associate-/l/ [=>] 98.5	\[ \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]
   associate-*l* [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   associate-*l* [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
   *-commutative [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]
   associate-*l/ [<=] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

3. Applied egg-rr 98.5%

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

   [Start] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   *-commutative [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot \sinh \left(\frac{1}{v}\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sinh-def [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\left(\left(v \cdot 2\right) \cdot v\right) \cdot \color{blue}{\frac{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}{2}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-*r/ [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{\left(\left(v \cdot 2\right) \cdot v\right) \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{2}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   *-commutative [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{\color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{2} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sinh-undef [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{\left(v \cdot \left(v \cdot 2\right)\right) \cdot \color{blue}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}}{2} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

4. Simplified 98.5%

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

   [Start] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{\left(v \cdot \left(v \cdot 2\right)\right) \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)}{2} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-/l* [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{v \cdot \left(v \cdot 2\right)}{\frac{2}{2 \cdot \sinh \left(\frac{1}{v}\right)}}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-/r* [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{v \cdot \left(v \cdot 2\right)}{\color{blue}{\frac{\frac{2}{2}}{\sinh \left(\frac{1}{v}\right)}}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   metadata-eval [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{v \cdot \left(v \cdot 2\right)}{\frac{\color{blue}{1}}{\sinh \left(\frac{1}{v}\right)}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-/l* [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{v}{\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{v \cdot 2}}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

5. Applied egg-rr 98.7%

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

   [Start] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\frac{v}{\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{v \cdot 2}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   frac-2neg [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{-v}{-\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{v \cdot 2}}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   div-inv [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\left(-v\right) \cdot \frac{1}{-\frac{\frac{1}{\sinh \left(\frac{1}{v}\right)}}{v \cdot 2}}\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-/l/ [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\frac{1}{\left(v \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)}}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-/r* [=>] 98.7	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\frac{\frac{1}{v \cdot 2}}{\sinh \left(\frac{1}{v}\right)}}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   *-commutative [=>] 98.7	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\left(-v\right) \cdot \frac{1}{-\frac{\frac{1}{\color{blue}{2 \cdot v}}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-/r* [=>] 98.7	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\left(-v\right) \cdot \frac{1}{-\frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   metadata-eval [=>] 98.7	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\left(-v\right) \cdot \frac{1}{-\frac{\frac{\color{blue}{0.5}}{v}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

6. Applied egg-rr 98.5%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{\frac{0.5}{v}} \cdot \sinh \left(\frac{1}{v}\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   Proof

   [Start] 98.7	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\left(-v\right) \cdot \frac{1}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   add-sqr-sqrt [=>] -0.0	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)} \cdot \frac{1}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sqrt-unprod [=>] 49.4	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}} \cdot \frac{1}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sqr-neg [=>] 49.4	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\sqrt{\color{blue}{v \cdot v}} \cdot \frac{1}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sqrt-unprod [<=] 49.4	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)} \cdot \frac{1}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   add-sqr-sqrt [<=] 49.4	\[ \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{v} \cdot \frac{1}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   expm1-log1p-u [=>] -0.0	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \frac{1}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   un-div-inv [=>] -0.0	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{v}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   add-sqr-sqrt [=>] -0.0	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sqrt-unprod [=>] -0.0	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{v \cdot v}}}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sqr-neg [<=] -0.0	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)}}}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   sqrt-unprod [<=] -0.0	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   add-sqr-sqrt [<=] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{-v}}{-\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   distribute-neg-frac [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-v}{\color{blue}{\frac{-\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-/r/ [=>] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-v}{-\frac{0.5}{v}} \cdot \sinh \left(\frac{1}{v}\right)}\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

7. Applied egg-rr 98.7%

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

   [Start] 98.5	\[ \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{v}{\frac{0.5}{v}} \cdot \sinh \left(\frac{1}{v}\right)\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   expm1-log1p-u [<=] 98.7	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\frac{v}{\frac{0.5}{v}} \cdot \sinh \left(\frac{1}{v}\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
   associate-*l/ [=>] 98.7	\[ \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\frac{v \cdot \sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

8. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	7008

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

Alternative 2
Accuracy	98.3%
Cost	3616

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

Alternative 3
Accuracy	98.2%
Cost	3616

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

Alternative 4
Accuracy	98.4%
Cost	3616

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

Alternative 5
Accuracy	59.9%
Cost	864

\[\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\left(\left(1 + \frac{1}{v}\right) + \frac{1}{v \cdot v} \cdot \left(0.5 + \frac{0.16666666666666666}{v}\right)\right) - -0.5} \]

Alternative 6
Accuracy	58.8%
Cost	608

\[\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{1.5 + \left(\frac{1}{v} + \frac{0.5}{v \cdot v}\right)} \]

Alternative 7
Accuracy	58.8%
Cost	608

\[\frac{\frac{cosTheta_i \cdot cosTheta_O}{v \cdot v}}{\left(\frac{1}{v} + 1.5\right) + \frac{0.5}{v \cdot v}} \]

Alternative 8
Accuracy	58.2%
Cost	288

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

Alternative 9
Accuracy	58.2%
Cost	288

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

Alternative 10
Accuracy	57.7%
Cost	224

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

Alternative 11
Accuracy	57.7%
Cost	224

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

Alternative 12
Accuracy	57.7%
Cost	224

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

Alternative 13
Accuracy	58.2%
Cost	224

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

Reproduce

herbie shell --seed 2023147 
(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)))