HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 20.1s

Alternatives: 14

Speedup: 1.5×

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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* (exp (/ (* sinTheta_i (- sinTheta_O)) v)) cosTheta_O_m)
    (* (/ cosTheta_i_m (* v (sinh (/ 1.0 v)))) (/ 0.5 v))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * ((exp(((sintheta_i * -sintheta_o) / v)) * costheta_o_m) * ((costheta_i_m / (v * sinh((1.0e0 / v)))) * (0.5e0 / v))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((exp(((sinTheta_i * -sinTheta_O) / v)) * cosTheta_O_m) * ((cosTheta_i_m / (v * sinh((single(1.0) / v)))) * (single(0.5) / v))));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. exp-lowering-exp.f32 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. neg-lowering-neg.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.7%

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

   1. associate-/r* N/A

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

   2. associate-/l/ N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

   7. *-lowering-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. /-lowering-/.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. sinh-lowering-sinh.f32 N/A

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

   12. /-lowering-/.f32 N/A

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

   13. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 98.7% accurate, 1.5× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (fma (- cosTheta_O_m) (/ (* sinTheta_i sinTheta_O) v) cosTheta_O_m)
    (* (/ cosTheta_i_m v) (/ (/ 0.5 v) (sinh (/ 1.0 v))))))))

C

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

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(fma(Float32(-cosTheta_O_m), Float32(Float32(sinTheta_i * sinTheta_O) / v), cosTheta_O_m) * Float32(Float32(cosTheta_i_m / v) * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v)))))))
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. exp-lowering-exp.f32 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. neg-lowering-neg.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.7%

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

   1. associate-/r* N/A

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

   2. associate-/l/ N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

   7. *-lowering-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. /-lowering-/.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. sinh-lowering-sinh.f32 N/A

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

   12. /-lowering-/.f32 N/A

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

   13. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. sinh-lowering-sinh.f32 N/A

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

   8. /-lowering-/.f32 98.8

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in sinTheta_i around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. associate-/l* N/A

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

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

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

    5. mul-1-neg N/A

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

    6. accelerator-lowering-fma.f32 N/A

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

    7. mul-1-neg N/A

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

    8. neg-lowering-neg.f32 N/A

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

    9. /-lowering-/.f32 N/A

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

    10. *-lowering-*.f32 98.8

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

    \[\leadsto \mathsf{fma}\left(-cosTheta\_O, \frac{sinTheta\_i \cdot sinTheta\_O}{v}, cosTheta\_O\right) \cdot \left(\frac{cosTheta\_i}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]
13. Add Preprocessing

Alternative 3: 98.6% accurate, 1.6× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* cosTheta_O_m (fma sinTheta_O (/ (- sinTheta_i) v) 1.0))
    (/ cosTheta_i_m (* (sinh (/ 1.0 v)) (* v (* v 2.0))))))))

C

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

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * fma(sinTheta_O, Float32(Float32(-sinTheta_i) / v), Float32(1.0))) * Float32(cosTheta_i_m / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0))))))))
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. exp-lowering-exp.f32 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. neg-lowering-neg.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

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

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

   5. mul-1-neg N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. mul-1-neg N/A

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

   10. /-lowering-/.f32 N/A

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

   11. mul-1-neg N/A

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

   12. neg-lowering-neg.f32 98.7

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 98.5% accurate, 1.8× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (* cosTheta_O_m (* (/ cosTheta_i_m (* v (sinh (/ 1.0 v)))) (/ 0.5 v))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * (costheta_o_m * ((costheta_i_m / (v * sinh((1.0e0 / v)))) * (0.5e0 / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(cosTheta_O_m * Float32(Float32(cosTheta_i_m / Float32(v * sinh(Float32(Float32(1.0) / v)))) * Float32(Float32(0.5) / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (cosTheta_O_m * ((cosTheta_i_m / (v * sinh((single(1.0) / v)))) * (single(0.5) / v))));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. exp-lowering-exp.f32 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. neg-lowering-neg.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.7%

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

   1. associate-/r* N/A

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

   2. associate-/l/ N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

   7. *-lowering-*.f32 N/A

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

   8. associate-/l/ N/A

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

   9. /-lowering-/.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. sinh-lowering-sinh.f32 N/A

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

   12. /-lowering-/.f32 N/A

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

   13. /-lowering-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.5%

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

Alternative 5: 98.4% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (* cosTheta_O_m (/ cosTheta_i_m (* (sinh (/ 1.0 v)) (* v (* v 2.0))))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * (costheta_o_m * (costheta_i_m / (sinh((1.0e0 / v)) * (v * (v * 2.0e0))))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(cosTheta_O_m * Float32(cosTheta_i_m / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0))))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (cosTheta_O_m * (cosTheta_i_m / (sinh((single(1.0) / v)) * (v * (v * single(2.0)))))));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. exp-lowering-exp.f32 N/A

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

   9. distribute-neg-frac2 N/A

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

   10. /-lowering-/.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. neg-lowering-neg.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 64.0% accurate, 2.9× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    cosTheta_O_m
    (/
     1.0
     (/
      (*
       v
       (fma
        2.0
        (fma
         sinTheta_i
         (/ sinTheta_O v)
         (/
          (fma
           (* sinTheta_O (* sinTheta_i sinTheta_i))
           (* sinTheta_O 0.5)
           0.16666666666666666)
          (* v v)))
        2.0))
      cosTheta_i_m))))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

7. Simplified 62.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

9. Applied egg-rr 62.4%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{1}{v \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right), 0.16666666666666666\right)}{v \cdot v}\right), 2\right)}\right)} \]
10. Step-by-step derivation

    1. un-div-inv N/A

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

    2. clear-num N/A

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

    3. /-lowering-/.f32 N/A

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

    4. /-lowering-/.f32 N/A

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

11. Applied egg-rr 62.4%

    \[\leadsto cosTheta\_O \cdot \color{blue}{\frac{1}{\frac{v \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right), 0.5 \cdot sinTheta\_O, 0.16666666666666666\right)}{v \cdot v}\right), 2\right)}{cosTheta\_i}}} \]

12. Final simplification 62.4%

    \[\leadsto cosTheta\_O \cdot \frac{1}{\frac{v \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right), sinTheta\_O \cdot 0.5, 0.16666666666666666\right)}{v \cdot v}\right), 2\right)}{cosTheta\_i}} \]
13. Add Preprocessing

Alternative 7: 64.0% accurate, 3.1× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (/
     (* cosTheta_O_m cosTheta_i_m)
     (fma
      2.0
      (fma
       sinTheta_i
       (/ sinTheta_O v)
       (/
        (fma
         0.5
         (* sinTheta_O (* sinTheta_O (* sinTheta_i sinTheta_i)))
         0.16666666666666666)
        (* v v)))
      2.0))
    v))))

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

7. Simplified 62.4%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f32 N/A

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

9. Applied egg-rr 62.4%

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

10. Final simplification 62.4%

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

Alternative 8: 64.0% accurate, 6.6× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* cosTheta_O_m cosTheta_i_m)
    (* v (+ 2.0 (/ 0.3333333333333333 (* v v))))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * ((costheta_o_m * costheta_i_m) / (v * (2.0e0 + (0.3333333333333333e0 / (v * v))))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * cosTheta_i_m) / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * cosTheta_i_m) / (v * (single(2.0) + (single(0.3333333333333333) / (v * v))))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

7. Simplified 62.4%

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

8. Taylor expanded in sinTheta_O around 0

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \cdot v} \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)} \cdot v} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}\right) \cdot v} \]

   3. metadata-eval N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}\right) \cdot v} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}\right) \cdot v} \]

   5. unpow2 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}\right) \cdot v} \]

   6. *-lowering-*.f32 62.4

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\left(2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}\right) \cdot v} \]

10. Simplified 62.4%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \cdot v} \]

11. Final simplification 62.4%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \]
12. Add Preprocessing

Alternative 9: 64.1% accurate, 6.6× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    cosTheta_O_m
    (/ cosTheta_i_m (* v (+ 2.0 (/ 0.3333333333333333 (* v v)))))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * (costheta_o_m * (costheta_i_m / (v * (2.0e0 + (0.3333333333333333e0 / (v * v)))))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(cosTheta_O_m * Float32(cosTheta_i_m / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (cosTheta_O_m * (cosTheta_i_m / (v * (single(2.0) + (single(0.3333333333333333) / (v * v)))))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

7. Simplified 62.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

9. Applied egg-rr 62.4%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{1}{v \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right), 0.16666666666666666\right)}{v \cdot v}\right), 2\right)}\right)} \]

10. Taylor expanded in sinTheta_i around 0

    \[\leadsto cosTheta\_O \cdot \color{blue}{\frac{cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]
11. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto cosTheta\_O \cdot \color{blue}{\frac{cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

    2. *-lowering-*.f32 N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

    3. +-lowering-+.f32 N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]

    4. associate-*r/ N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(2 + \color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}}\right)} \]

    5. metadata-eval N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(2 + \frac{\color{blue}{\frac{1}{3}}}{{v}^{2}}\right)} \]

    6. /-lowering-/.f32 N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(2 + \color{blue}{\frac{\frac{1}{3}}{{v}^{2}}}\right)} \]

    7. unpow2 N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}\right)} \]

    8. *-lowering-*.f32 62.4

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}\right)} \]

12. Simplified 62.4%

    \[\leadsto cosTheta\_O \cdot \color{blue}{\frac{cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}} \]
13. Add Preprocessing

Alternative 10: 58.8% accurate, 8.2× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_i_s (/ 1.0 (/ v (* cosTheta_O_m (* cosTheta_i_m 0.5)))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * (1.0e0 / (v / (costheta_o_m * (costheta_i_m * 0.5e0)))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (single(1.0) / (v / (cosTheta_O_m * (cosTheta_i_m * single(0.5))))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 56.6

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

5. Simplified 56.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f32 N/A

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

   3. /-lowering-/.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 57.1

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

7. Applied egg-rr 57.1%

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

Alternative 11: 58.8% accurate, 9.7× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (/ 0.5 (/ v (* cosTheta_O_m cosTheta_i_m))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * (0.5e0 / (v / (costheta_o_m * costheta_i_m))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (single(0.5) / (v / (cosTheta_O_m * cosTheta_i_m))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 56.6

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

5. Simplified 56.6%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. /-lowering-/.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f32 57.1

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

7. Applied egg-rr 57.1%

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

Alternative 12: 58.3% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (/ (* 0.5 (* cosTheta_O_m cosTheta_i_m)) v))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * ((0.5e0 * (costheta_o_m * costheta_i_m)) / v))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((single(0.5) * (cosTheta_O_m * cosTheta_i_m)) / v));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 56.6

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

5. Simplified 56.6%

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

6. Final simplification 56.6%

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

Alternative 13: 58.3% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* (/ 0.5 v) (* cosTheta_O_m cosTheta_i_m)))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * ((0.5e0 / v) * (costheta_o_m * costheta_i_m)))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((single(0.5) / v) * (cosTheta_O_m * cosTheta_i_m)));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 56.6

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

5. Simplified 56.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 56.5

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

7. Applied egg-rr 56.5%

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

Alternative 14: 58.3% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* cosTheta_O_m (* cosTheta_i_m (/ 0.5 v))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
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_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_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_o_s * (costheta_i_s * (costheta_o_m * (costheta_i_m * (0.5e0 / v))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * 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_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (cosTheta_O_m * (cosTheta_i_m * (single(0.5) / v))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

7. Simplified 62.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

9. Applied egg-rr 62.4%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{1}{v \cdot \mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{\mathsf{fma}\left(0.5, sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right), 0.16666666666666666\right)}{v \cdot v}\right), 2\right)}\right)} \]

10. Taylor expanded in v around inf

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

    1. /-lowering-/.f32 56.5

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

12. Simplified 56.5%

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

Reproduce

herbie shell --seed 2024199 
(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)))