HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 20.8s

Alternatives: 20

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

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)))

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);
}

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

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

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

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)))

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);
}

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

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

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

Alternative 1: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot cosTheta\_O}{\frac{-1}{\frac{\frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}}{v}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot -2\right)}}} \end{array} \]

FPCore

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

C

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

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 = (costheta_i * costheta_o) / ((-1.0e0) / ((exp(((sintheta_i * sintheta_o) / -v)) / v) / (v * (sinh((1.0e0 / v)) * (-2.0e0)))))
end function

Julia

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

MATLAB

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

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   2. lift-sinh.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   3. *-commutative N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   4. associate-*l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)} \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot e^{\frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}}\right) \cdot v} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot v} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot \color{blue}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot v} \]

   9. associate-*l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right)} \cdot v} \]

   10. remove-double-div N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \color{blue}{\frac{1}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}}\right)\right) \cdot v} \]

   11. lift-exp.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{\frac{1}{\color{blue}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}}\right)\right) \cdot v} \]

   12. exp-neg N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}}}\right)\right) \cdot v} \]

   13. lift-/.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)}}\right)\right) \cdot v} \]

   14. distribute-frac-neg2 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

   15. lift-neg.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

   16. lift-/.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

   17. lift-exp.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{\color{blue}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

6. Applied egg-rr 99.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\frac{2 \cdot \sinh \left(\frac{1}{v}\right)}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{v}}} \cdot v} \]

8. Applied egg-rr 99.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\frac{1}{\frac{\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{-v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot -2\right) \cdot v}}}} \]

9. Final simplification 99.0%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\frac{-1}{\frac{\frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}}}{v}}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot -2\right)}}} \]
10. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{v}}} \end{array} \]

FPCore

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

C

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)) * 2.0f) / (expf((sinTheta_i * (sinTheta_O / -v))) / 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 = (costheta_i * costheta_o) / (v * ((sinh((1.0e0 / v)) * 2.0e0) / (exp((sintheta_i * (sintheta_o / -v))) / v)))
end function

Julia

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

MATLAB

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

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   2. lift-sinh.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   3. *-commutative N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   4. associate-*l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)} \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right) \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) \cdot v} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot e^{\frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}}\right) \cdot v} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot v} \]

   8. lift-exp.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot \color{blue}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot v} \]

   9. associate-*l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right)} \cdot v} \]

   10. remove-double-div N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \color{blue}{\frac{1}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}}\right)\right) \cdot v} \]

   11. lift-exp.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{\frac{1}{\color{blue}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}}\right)\right) \cdot v} \]

   12. exp-neg N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}}}\right)\right) \cdot v} \]

   13. lift-/.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\mathsf{neg}\left(\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)}}\right)\right) \cdot v} \]

   14. distribute-frac-neg2 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

   15. lift-neg.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\color{blue}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

   16. lift-/.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

   17. lift-exp.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \left(v \cdot \frac{1}{\color{blue}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{\mathsf{neg}\left(v\right)}}}}\right)\right) \cdot v} \]

6. Applied egg-rr 98.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\frac{2 \cdot \sinh \left(\frac{1}{v}\right)}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{v}}} \cdot v} \]

7. Final simplification 98.7%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}}}{v}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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)))