HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 14.3s

Alternatives: 10

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((single(1.0) / sinh((single(-1.0) / v))) * (((single(-0.5) / v) * cosTheta_O_m) / ((exp(sinTheta_i) ^ (sinTheta_O / v)) * v))) * 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. 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 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. times-frac N/A

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

   8. associate-*l/ N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

8. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

10. Applied rewrites 98.8%

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

11. Final simplification 98.8%

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

Alternative 2: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \left(\left(\frac{cosTheta\_i\_m}{\sinh \left(\frac{-1}{v}\right)} \cdot \frac{cosTheta\_O\_m}{v}\right) \cdot \frac{\frac{sinTheta\_i \cdot sinTheta\_i}{v} \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot 0.5}{v} + \left(1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{-2 \cdot v}\right)\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (*
    (* (/ cosTheta_i_m (sinh (/ -1.0 v))) (/ cosTheta_O_m v))
    (/
     (+
      (*
       (/ (* sinTheta_i sinTheta_i) v)
       (/ (* (* sinTheta_O sinTheta_O) 0.5) v))
      (- 1.0 (/ (* sinTheta_O sinTheta_i) v)))
     (* -2.0 v))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * (((cosTheta_i_m / sinhf((-1.0f / v))) * (cosTheta_O_m / v)) * (((((sinTheta_i * sinTheta_i) / v) * (((sinTheta_O * sinTheta_O) * 0.5f) / v)) + (1.0f - ((sinTheta_O * sinTheta_i) / v))) / (-2.0f * v))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (costheta_o_s * (((costheta_i_m / sinh(((-1.0e0) / v))) * (costheta_o_m / v)) * (((((sintheta_i * sintheta_i) / v) * (((sintheta_o * sintheta_o) * 0.5e0) / v)) + (1.0e0 - ((sintheta_o * sintheta_i) / v))) / ((-2.0e0) * v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_i_m / sinh(Float32(Float32(-1.0) / v))) * Float32(cosTheta_O_m / v)) * Float32(Float32(Float32(Float32(Float32(sinTheta_i * sinTheta_i) / v) * Float32(Float32(Float32(sinTheta_O * sinTheta_O) * Float32(0.5)) / v)) + Float32(Float32(1.0) - Float32(Float32(sinTheta_O * sinTheta_i) / v))) / Float32(Float32(-2.0) * v)))))
end

MATLAB

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

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

   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 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in v around inf

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 N/A

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

   18. lower-/.f32 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f32 98.8

       \[\leadsto \frac{\left(1 - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v} \cdot \frac{\color{blue}{sinTheta\_i \cdot sinTheta\_i}}{v}}{-2 \cdot v} \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{cosTheta\_i}{\sinh \left(\frac{-1}{v}\right)}\right) \]

9. Applied rewrites 98.8%

   \[\leadsto \frac{\color{blue}{\left(1 - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v} \cdot \frac{sinTheta\_i \cdot sinTheta\_i}{v}}}{-2 \cdot v} \cdot \left(\frac{cosTheta\_O}{v} \cdot \frac{cosTheta\_i}{\sinh \left(\frac{-1}{v}\right)}\right) \]

10. Final simplification 98.8%

    \[\leadsto \left(\frac{cosTheta\_i}{\sinh \left(\frac{-1}{v}\right)} \cdot \frac{cosTheta\_O}{v}\right) \cdot \frac{\frac{sinTheta\_i \cdot sinTheta\_i}{v} \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot 0.5}{v} + \left(1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{-2 \cdot v} \]
11. Add Preprocessing

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((single(1.0) - ((sinTheta_O * sinTheta_i) / v)) / (single(-2.0) * v)) * ((cosTheta_i_m / sinh((single(-1.0) / v))) * (cosTheta_O_m / 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 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 98.8

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

9. Applied rewrites 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((single(-0.5) / v) * ((cosTheta_i_m / sinh((single(-1.0) / v))) * (cosTheta_O_m / 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 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l* N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 5: 55.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \frac{cosTheta\_O\_m \cdot cosTheta\_i\_m}{\left(2 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, \frac{\mathsf{fma}\left(\left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_O, 0.5, 0.16666666666666666\right)}{v}\right) \cdot -2}{v}\right) \cdot v}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (* cosTheta_O_m cosTheta_i_m)
    (*
     (-
      2.0
      (/
       (*
        (fma
         sinTheta_i
         sinTheta_O
         (/
          (fma
           (* (* (* sinTheta_i sinTheta_i) sinTheta_O) sinTheta_O)
           0.5
           0.16666666666666666)
          v))
        -2.0)
       v))
     v)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * ((cosTheta_O_m * cosTheta_i_m) / ((2.0f - ((fmaf(sinTheta_i, sinTheta_O, (fmaf((((sinTheta_i * sinTheta_i) * sinTheta_O) * sinTheta_O), 0.5f, 0.16666666666666666f) / v)) * -2.0f) / v)) * v)));
}

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(cosTheta_O_m * cosTheta_i_m) / Float32(Float32(Float32(2.0) - Float32(Float32(fma(sinTheta_i, sinTheta_O, Float32(fma(Float32(Float32(Float32(sinTheta_i * sinTheta_i) * sinTheta_O) * sinTheta_O), Float32(0.5), Float32(0.16666666666666666)) / v)) * Float32(-2.0)) / v)) * v))))
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
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 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. exp-neg N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l/ N/A

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

   13. *-lft-identity N/A

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

   14. associate-/l/ N/A

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in v around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-neg.f32 N/A

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

   5. lower--.f32 N/A

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

9. Applied rewrites 64.0%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{\left(-v\right) \cdot \left(\frac{-2 \cdot \mathsf{fma}\left(sinTheta\_i, sinTheta\_O, \frac{\mathsf{fma}\left(\left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_O, 0.5, 0.16666666666666666\right)}{v}\right)}{v} - 2\right)}} \]

10. Final simplification 64.0%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(2 - \frac{\mathsf{fma}\left(sinTheta\_i, sinTheta\_O, \frac{\mathsf{fma}\left(\left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_O, 0.5, 0.16666666666666666\right)}{v}\right) \cdot -2}{v}\right) \cdot v} \]
11. Add Preprocessing

Alternative 6: 59.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (single(0.5) / ((v / cosTheta_i_m) / cosTheta_O_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. 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 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. 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 58.4

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

7. Applied rewrites 58.4%

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

9. Applied rewrites 58.5%

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

Alternative 7: 59.3% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (single(1.0) / (v / ((cosTheta_O_m * 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. 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 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. 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 58.4

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

7. Applied rewrites 58.4%

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

9. Applied rewrites 58.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{v}{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}}} \]
10. Add Preprocessing

Alternative 8: 58.7% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((single(-0.5) * cosTheta_O_m) * cosTheta_i_m) / -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 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around -inf

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

7. Applied rewrites 56.1%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) - \frac{\mathsf{fma}\left(-0.5 \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot cosTheta\_i\right), cosTheta\_O, \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \mathsf{fma}\left(\left(\left(sinTheta\_i \cdot sinTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_O, 0.5, -0.16666666666666666\right)}{v} \cdot 0.5\right)}{v}}{-v}} \]

8. Taylor expanded in v around inf

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

10. Applied rewrites 58.4%

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

Alternative 9: 58.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((cosTheta_O_m * 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. 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 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. 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 58.4

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

7. Applied rewrites 58.4%

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

9. Applied rewrites 58.4%

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

Alternative 10: 58.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((cosTheta_O_m * 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. 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 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. 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 58.4

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

7. Applied rewrites 58.4%

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

8. Final simplification 58.4%

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

Reproduce

herbie shell --seed 2024304 
(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)))