HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 15.1s

Alternatives: 16

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: 98.5% 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: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. /-rgt-identity N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

   11. unpow-1 N/A

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

   12. lift-pow.f32 N/A

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

   13. lower-*.f32 98.7

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

   14. lift-pow.f32 N/A

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

   15. unpow-1 N/A

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

   16. lift-/.f32 N/A

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

   17. clear-num N/A

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

   18. /-rgt-identity 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\_m\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (*
    (* (* cosTheta_O cosTheta_i_m) (/ 1.0 v))
    (exp (/ (* sinTheta_O sinTheta_i) (- v))))
   (/ (sinh (/ 1.0 v)) (/ 0.5 v)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) * (1.0f / v)) * expf(((sinTheta_O * sinTheta_i) / -v))) / (sinhf((1.0f / v)) / (0.5f / v)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((((costheta_o * costheta_i_m) * (1.0e0 / v)) * exp(((sintheta_o * sintheta_i) / -v))) / (sinh((1.0e0 / v)) / (0.5e0 / v)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i_m) * Float32(Float32(1.0) / v)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) * (single(1.0) / v)) * exp(((sinTheta_O * sinTheta_i) / -v))) / (sinh((single(1.0) / v)) / (single(0.5) / 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
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. /-rgt-identity N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

   11. unpow-1 N/A

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

   12. lift-pow.f32 N/A

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

   13. lower-*.f32 98.7

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

   14. lift-pow.f32 N/A

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

   15. unpow-1 N/A

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

   16. lift-/.f32 N/A

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

   17. clear-num N/A

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

   18. /-rgt-identity 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. lift-sinh.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. sinh-neg N/A

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

   13. lift-sinh.f32 N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg N/A

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

   16. /-rgt-identity N/A

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

   17. div-inv N/A

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

   18. lift-/.f32 N/A

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

   19. remove-double-div N/A

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

   20. *-commutative N/A

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

   21. metadata-eval N/A

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

   22. div-inv N/A

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

   23. clear-num N/A

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

8. Applied rewrites 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}} \]

9. Final simplification 98.9%

   \[\leadsto \frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \]
10. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. /-rgt-identity N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

   11. unpow-1 N/A

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

   12. lift-pow.f32 N/A

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

   13. lower-*.f32 98.7

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

   14. lift-pow.f32 N/A

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

   15. unpow-1 N/A

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

   16. lift-/.f32 N/A

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

   17. clear-num N/A

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

   18. /-rgt-identity 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 98.8

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

8. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/l* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

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

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

   14. lift-sinh.f32 N/A

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

   15. sinh-neg N/A

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

   16. lift-/.f32 N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. lift-/.f32 N/A

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

   20. lift-sinh.f32 N/A

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 63.6

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

7. Applied rewrites 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]

8. Taylor expanded in sinTheta_i around 0

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

   1. mul-1-neg N/A

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

   2. times-frac N/A

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

   3. associate-*r/ N/A

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

   4. distribute-neg-frac2 N/A

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

   5. neg-sub0 N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{\color{blue}{0 - \left(e^{\frac{-1}{v}} - \frac{1}{e^{\frac{-1}{v}}}\right)}} \]

   6. rec-exp N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{0 - \left(e^{\frac{-1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{-1}{v}\right)}}\right)} \]

   7. distribute-neg-frac N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{0 - \left(e^{\frac{-1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{v}}}\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{0 - \left(e^{\frac{-1}{v}} - e^{\frac{\color{blue}{1}}{v}}\right)} \]

   9. associate-+l- N/A

      \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{\color{blue}{\left(0 - e^{\frac{-1}{v}}\right) + e^{\frac{1}{v}}}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{\left(0 - e^{\frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}}\right) + e^{\frac{1}{v}}} \]

   11. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{\left(0 - e^{\color{blue}{\mathsf{neg}\left(\frac{1}{v}\right)}}\right) + e^{\frac{1}{v}}} \]

   12. rec-exp N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{\left(0 - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}\right) + e^{\frac{1}{v}}} \]

   13. neg-sub0 N/A

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

   14. +-commutative N/A

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

10. Applied rewrites 98.3%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
11. Add Preprocessing

Alternative 6: 69.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_O \cdot cosTheta\_i\_m}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{v}{\frac{0.5}{\frac{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v}}{v} - -1}{v}}}} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (*
    (/ (* cosTheta_O cosTheta_i_m) v)
    (exp (/ (* sinTheta_O sinTheta_i) (- v))))
   (/
    v
    (/
     0.5
     (/
      (-
       (/ (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) v) v)
       -1.0)
      v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / (v / (0.5f / (((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / v) / v) - -1.0f) / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((((costheta_o * costheta_i_m) / v) * exp(((sintheta_o * sintheta_i) / -v))) / (v / (0.5e0 / (((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / v) / v) - (-1.0e0)) / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i_m) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(v / Float32(Float32(0.5) / Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / v) / v) - Float32(-1.0)) / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / (v / (single(0.5) / (((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / v) / v) - single(-1.0)) / 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
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{v}{\frac{0.5}{\sinh \left(\frac{1}{v}\right)}}}} \]

7. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{\frac{1}{2}}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}}}}} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{\frac{1}{2}}{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}}}} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{\frac{1}{2}}{\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}}}}} \]

   3. neg-mul-1 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{\frac{1}{2}}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{-1 \cdot v}}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{\frac{1}{2}}{\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}}}}} \]

9. Applied rewrites 69.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{v}{\frac{0.5}{\color{blue}{\frac{-1 - \frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v}}{v}}{-v}}}}} \]

10. Final simplification 69.6%

    \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{v}{\frac{0.5}{\frac{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v}}{v} - -1}{v}}}} \]
11. Add Preprocessing

Alternative 7: 69.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_O \cdot cosTheta\_i\_m}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v}}{v} - -1}{v} \cdot 2\right) \cdot v} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (*
    (/ (* cosTheta_O cosTheta_i_m) v)
    (exp (/ (* sinTheta_O sinTheta_i) (- v))))
   (*
    (*
     (/
      (-
       (/ (/ (- (/ 0.008333333333333333 (* v v)) -0.16666666666666666) v) v)
       -1.0)
      v)
     2.0)
    v))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((((((((0.008333333333333333f / (v * v)) - -0.16666666666666666f) / v) / v) - -1.0f) / v) * 2.0f) * v));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((((costheta_o * costheta_i_m) / v) * exp(((sintheta_o * sintheta_i) / -v))) / ((((((((0.008333333333333333e0 / (v * v)) - (-0.16666666666666666e0)) / v) / v) - (-1.0e0)) / v) * 2.0e0) * v))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i_m) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.008333333333333333) / Float32(v * v)) - Float32(-0.16666666666666666)) / v) / v) - Float32(-1.0)) / v) * Float32(2.0)) * v)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((((((((single(0.008333333333333333) / (v * v)) - single(-0.16666666666666666)) / v) / v) - single(-1.0)) / v) * single(2.0)) * 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

3. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}{v}} \cdot 2\right) \cdot v} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)\right)}}{v} \cdot 2\right) \cdot v} \]

   3. sub-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{v} \cdot 2\right) \cdot v} \]

   4. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\left(\color{blue}{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} \cdot -1} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{v} \cdot 2\right) \cdot v} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} \cdot -1 + \color{blue}{-1}\right)\right)}{v} \cdot 2\right) \cdot v} \]

   6. distribute-lft1-in N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1\right) \cdot -1}\right)}{v} \cdot 2\right) \cdot v} \]

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

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1\right) \cdot \left(\mathsf{neg}\left(-1\right)\right)}}{v} \cdot 2\right) \cdot v} \]

   8. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1\right) \cdot \color{blue}{1}}{v} \cdot 2\right) \cdot v} \]

   9. *-rgt-identity N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1}}{v} \cdot 2\right) \cdot v} \]

   10. lower-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1}{v}} \cdot 2\right) \cdot v} \]

5. Applied rewrites 69.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v}}{v} - -1}{v}} \cdot 2\right) \cdot v} \]

6. Final simplification 69.6%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v}}{v} - -1}{v} \cdot 2\right) \cdot v} \]
7. Add Preprocessing

Alternative 8: 63.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_O \cdot cosTheta\_i\_m}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{0.16666666666666666}{v \cdot v} - -1}{v} \cdot 2\right) \cdot v} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (*
    (/ (* cosTheta_O cosTheta_i_m) v)
    (exp (/ (* sinTheta_O sinTheta_i) (- v))))
   (* (* (/ (- (/ 0.16666666666666666 (* v v)) -1.0) v) 2.0) v))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / (((((0.16666666666666666f / (v * v)) - -1.0f) / v) * 2.0f) * v));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((((costheta_o * costheta_i_m) / v) * exp(((sintheta_o * sintheta_i) / -v))) / (((((0.16666666666666666e0 / (v * v)) - (-1.0e0)) / v) * 2.0e0) * v))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i_m) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) - Float32(-1.0)) / v) * Float32(2.0)) * v)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / (((((single(0.16666666666666666) / (v * v)) - single(-1.0)) / v) * single(2.0)) * 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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}} \cdot 2\right) \cdot v} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}} \cdot 2\right) \cdot v} \]

   2. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1}}{v} \cdot 2\right) \cdot v} \]

   3. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}}{v} \cdot 2\right) \cdot v} \]

   4. sub-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} - -1}}{v} \cdot 2\right) \cdot v} \]

   5. lower--.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} - -1}}{v} \cdot 2\right) \cdot v} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\frac{\frac{1}{6} \cdot 1}{{v}^{2}}} - -1}{v} \cdot 2\right) \cdot v} \]

   7. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\frac{\color{blue}{\frac{1}{6}}}{{v}^{2}} - -1}{v} \cdot 2\right) \cdot v} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\frac{\frac{1}{6}}{{v}^{2}}} - -1}{v} \cdot 2\right) \cdot v} \]

   9. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\frac{\frac{1}{6}}{\color{blue}{v \cdot v}} - -1}{v} \cdot 2\right) \cdot v} \]

   10. lower-*.f32 63.6

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\frac{0.16666666666666666}{\color{blue}{v \cdot v}} - -1}{v} \cdot 2\right) \cdot v} \]

5. Applied rewrites 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{\frac{0.16666666666666666}{v \cdot v} - -1}{v}} \cdot 2\right) \cdot v} \]

6. Final simplification 63.6%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{0.16666666666666666}{v \cdot v} - -1}{v} \cdot 2\right) \cdot v} \]
7. Add Preprocessing

Alternative 9: 63.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{\frac{cosTheta\_O \cdot cosTheta\_i\_m}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (*
    (/ (* cosTheta_O cosTheta_i_m) v)
    (exp (/ (* sinTheta_O sinTheta_i) (- v))))
   (+ (/ 0.3333333333333333 (* v v)) 2.0))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((0.3333333333333333f / (v * v)) + 2.0f));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((((costheta_o * costheta_i_m) / v) * exp(((sintheta_o * sintheta_i) / -v))) / ((0.3333333333333333e0 / (v * v)) + 2.0e0))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(cosTheta_O * cosTheta_i_m) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((((cosTheta_O * cosTheta_i_m) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(0.3333333333333333) / (v * 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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 63.6

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

5. Applied rewrites 63.6%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]

6. Final simplification 63.6%

   \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
7. Add Preprocessing

Alternative 10: 58.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(\frac{\frac{1}{\frac{\frac{1}{cosTheta\_O}}{cosTheta\_i\_m}}}{v} \cdot 0.5\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* (/ (/ 1.0 (/ (/ 1.0 cosTheta_O) cosTheta_i_m)) v) 0.5)))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((1.0f / ((1.0f / cosTheta_O) / cosTheta_i_m)) / v) * 0.5f);
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((1.0e0 / ((1.0e0 / costheta_o) / costheta_i_m)) / v) * 0.5e0)
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / cosTheta_O) / cosTheta_i_m)) / v) * Float32(0.5)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((single(1.0) / ((single(1.0) / cosTheta_O) / cosTheta_i_m)) / v) * single(0.5));
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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 57.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. lower-*.f32 57.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v} \]

8. Applied rewrites 57.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.3%

    \[\leadsto 0.5 \cdot \frac{\frac{1}{\frac{\frac{1}{cosTheta\_O}}{cosTheta\_i}}}{v} \]

11. Final simplification 58.3%

    \[\leadsto \frac{\frac{1}{\frac{\frac{1}{cosTheta\_O}}{cosTheta\_i}}}{v} \cdot 0.5 \]
12. Add Preprocessing

Alternative 11: 58.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{\frac{0.5}{v}}{\frac{\frac{1}{cosTheta\_O}}{cosTheta\_i\_m}} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (/ (/ 0.5 v) (/ (/ 1.0 cosTheta_O) cosTheta_i_m))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((0.5f / v) / ((1.0f / cosTheta_O) / cosTheta_i_m));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((0.5e0 / v) / ((1.0e0 / costheta_o) / costheta_i_m))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(1.0) / cosTheta_O) / cosTheta_i_m)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((single(0.5) / v) / ((single(1.0) / cosTheta_O) / cosTheta_i_m));
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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 57.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. lower-*.f32 57.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v} \]

8. Applied rewrites 57.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.3%

    \[\leadsto \frac{\frac{0.5}{v}}{\color{blue}{\frac{\frac{1}{cosTheta\_O}}{cosTheta\_i}}} \]
11. Add Preprocessing

Alternative 12: 58.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 57.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. lower-*.f32 57.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v} \]

8. Applied rewrites 57.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot v}{cosTheta\_i \cdot cosTheta\_O}}} \]

11. Final simplification 58.2%

    \[\leadsto \frac{1}{\frac{2 \cdot v}{cosTheta\_O \cdot cosTheta\_i}} \]
12. Add Preprocessing

Alternative 13: 58.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{0.5}{\frac{v}{cosTheta\_O \cdot cosTheta\_i\_m}} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (/ 0.5 (/ v (* cosTheta_O cosTheta_i_m)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (0.5f / (v / (cosTheta_O * cosTheta_i_m)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 / (v / (costheta_o * costheta_i_m)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(0.5) / Float32(v / Float32(cosTheta_O * cosTheta_i_m))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (single(0.5) / (v / (cosTheta_O * cosTheta_i_m)));
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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 57.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. lower-*.f32 57.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v} \]

8. Applied rewrites 57.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.2%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

11. Final simplification 58.2%

    \[\leadsto \frac{0.5}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \]
12. Add Preprocessing

Alternative 14: 58.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{\left(cosTheta\_O \cdot cosTheta\_i\_m\right) \cdot 0.5}{v} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (/ (* (* cosTheta_O cosTheta_i_m) 0.5) v)))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((cosTheta_O * cosTheta_i_m) * 0.5f) / v);
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((costheta_o * costheta_i_m) * 0.5e0) / v)
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_O * cosTheta_i_m) * Float32(0.5)) / v))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((cosTheta_O * cosTheta_i_m) * single(0.5)) / 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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 57.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. lower-*.f32 57.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v} \]

8. Applied rewrites 57.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 57.9%

    \[\leadsto \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{\color{blue}{v}} \]

11. Final simplification 57.9%

    \[\leadsto \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v} \]
12. Add Preprocessing

Alternative 15: 58.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(\left(cosTheta\_O \cdot cosTheta\_i\_m\right) \cdot \frac{0.5}{v}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* (* cosTheta_O cosTheta_i_m) (/ 0.5 v))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((cosTheta_O * cosTheta_i_m) * (0.5f / v));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((costheta_o * costheta_i_m) * (0.5e0 / v))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(cosTheta_O * cosTheta_i_m) * Float32(Float32(0.5) / v)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((cosTheta_O * cosTheta_i_m) * (single(0.5) / 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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 57.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. lower-*.f32 57.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v} \]

8. Applied rewrites 57.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 57.9%

    \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)} \]

11. Final simplification 57.9%

    \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{0.5}{v} \]
12. Add Preprocessing

Alternative 16: 58.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\_m\right) \cdot 0.5\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* (* (/ cosTheta_O v) cosTheta_i_m) 0.5)))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((cosTheta_O / v) * cosTheta_i_m) * 0.5f);
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((costheta_o / v) * costheta_i_m) * 0.5e0)
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_O / v) * cosTheta_i_m) * Float32(0.5)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((cosTheta_O / v) * cosTheta_i_m) * single(0.5));
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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 57.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. lower-*.f32 57.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v} \]

8. Applied rewrites 57.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 57.9%

    \[\leadsto \left(\frac{0.5}{v} \cdot cosTheta\_O\right) \cdot \color{blue}{cosTheta\_i} \]
11. Step-by-step derivation

12. Applied rewrites 57.9%

    \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)} \]

13. Final simplification 57.9%

    \[\leadsto \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot 0.5 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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)))