HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.9%

Time: 21.4s

Alternatives: 19

Speedup: 1.8×

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: 98.5% 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: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \frac{\left(\frac{e^{sinTheta\_O \cdot \frac{-sinTheta\_i}{v}}}{v} \cdot cosTheta\_i\_m\right) \cdot cosTheta\_O\_m}{\frac{\sinh \left(\frac{-1}{v}\right) \cdot -2}{\frac{1}{v}}}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (* (/ (exp (* sinTheta_O (/ (- sinTheta_i) v))) v) cosTheta_i_m)
     cosTheta_O_m)
    (/ (* (sinh (/ -1.0 v)) -2.0) (/ 1.0 v))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * ((((expf((sinTheta_O * (-sinTheta_i / v))) / v) * cosTheta_i_m) * cosTheta_O_m) / ((sinhf((-1.0f / v)) * -2.0f) / (1.0f / v))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (costheta_o_s * ((((exp((sintheta_o * (-sintheta_i / v))) / v) * costheta_i_m) * costheta_o_m) / ((sinh(((-1.0e0) / v)) * (-2.0e0)) / (1.0e0 / v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(Float32(exp(Float32(sinTheta_O * Float32(Float32(-sinTheta_i) / v))) / v) * cosTheta_i_m) * cosTheta_O_m) / Float32(Float32(sinh(Float32(Float32(-1.0) / v)) * Float32(-2.0)) / Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((((exp((sinTheta_O * (-sinTheta_i / v))) / v) * cosTheta_i_m) * cosTheta_O_m) / ((sinh((single(-1.0) / v)) * single(-2.0)) / (single(1.0) / v))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 99.1

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. remove-double-neg N/A

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

   9. lift-sinh.f32 N/A

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

   10. sinh-neg N/A

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

   11. neg-mul-1 N/A

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

   12. lift-/.f32 N/A

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

   13. div-inv N/A

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

   14. lift-/.f32 N/A

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

   15. lift-sinh.f32 N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. distribute-lft-neg-in N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f32 99.1

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

6. Applied rewrites 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \frac{\left(cosTheta\_i\_m \cdot cosTheta\_O\_m\right) \cdot \frac{1}{v}}{\frac{\sinh \left(\frac{-1}{v}\right) \cdot -2}{\frac{1}{v}}}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (* (* cosTheta_i_m cosTheta_O_m) (/ 1.0 v))
    (/ (* (sinh (/ -1.0 v)) -2.0) (/ 1.0 v))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * (((cosTheta_i_m * cosTheta_O_m) * (1.0f / v)) / ((sinhf((-1.0f / v)) * -2.0f) / (1.0f / v))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (costheta_o_s * (((costheta_i_m * costheta_o_m) * (1.0e0 / v)) / ((sinh(((-1.0e0) / v)) * (-2.0e0)) / (1.0e0 / v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_i_m * cosTheta_O_m) * Float32(Float32(1.0) / v)) / Float32(Float32(sinh(Float32(Float32(-1.0) / v)) * Float32(-2.0)) / Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((cosTheta_i_m * cosTheta_O_m) * (single(1.0) / v)) / ((sinh((single(-1.0) / v)) * single(-2.0)) / (single(1.0) / v))));
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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. remove-double-neg N/A

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

   9. lift-sinh.f32 N/A

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

   10. sinh-neg N/A

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

   11. neg-mul-1 N/A

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

   12. lift-/.f32 N/A

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

   13. div-inv N/A

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

   14. lift-/.f32 N/A

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

   15. lift-sinh.f32 N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. distribute-lft-neg-in N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 98.8

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 98.8

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.8

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

8. Applied rewrites 98.8%

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

9. Taylor expanded in sinTheta_O around 0

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

    1. lower-/.f32 98.6

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

11. Applied rewrites 98.6%

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

12. Final simplification 98.6%

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

Reproduce

herbie shell --seed 2024234 
(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)))