HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.9%

Time: 23.9s

Alternatives: 19

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.7× speedup

Math

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

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
   (*
    (*
     (/
      (+
       (/ 1.0 v)
       (*
        sinTheta_i
        (/
         (- (* (/ 0.5 v) (* sinTheta_i (* sinTheta_O sinTheta_O))) sinTheta_O)
         (* v v))))
      (/ (sinh (/ 1.0 v)) (/ 0.5 v)))
     cosTheta_i_m)
    cosTheta_O_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 * (((((1.0f / v) + (sinTheta_i * ((((0.5f / v) * (sinTheta_i * (sinTheta_O * sinTheta_O))) - sinTheta_O) / (v * v)))) / (sinhf((1.0f / v)) / (0.5f / v))) * cosTheta_i_m) * cosTheta_O_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 * (((((1.0e0 / v) + (sintheta_i * ((((0.5e0 / v) * (sintheta_i * (sintheta_o * sintheta_o))) - sintheta_o) / (v * v)))) / (sinh((1.0e0 / v)) / (0.5e0 / v))) * costheta_i_m) * costheta_o_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(Float32(Float32(Float32(1.0) / v) + Float32(sinTheta_i * Float32(Float32(Float32(Float32(Float32(0.5) / v) * Float32(sinTheta_i * Float32(sinTheta_O * sinTheta_O))) - sinTheta_O) / Float32(v * v)))) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v))) * cosTheta_i_m) * cosTheta_O_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(1.0) / v) + (sinTheta_i * ((((single(0.5) / v) * (sinTheta_i * (sinTheta_O * sinTheta_O))) - sinTheta_O) / (v * v)))) / (sinh((single(1.0) / v)) / (single(0.5) / v))) * cosTheta_i_m) * cosTheta_O_m));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\color{blue}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{e^{0 - \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\color{blue}{\left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right) + \frac{1}{v}\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{v} + sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{v}\right), \left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} + -1 \cdot \frac{sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} + \left(\mathsf{neg}\left(\frac{sinTheta\_O}{{v}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} - \frac{sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right), \left(\frac{sinTheta\_O}{{v}^{2}}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

7. Simplified 98.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\left(\frac{1}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{2}}{v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v} - \frac{sinTheta\_O}{v \cdot v}\right)\right) \cdot \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot cosTheta\_i\right), \color{blue}{cosTheta\_O}\right) \]

9. Applied egg-rr 98.9%

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

Alternative 2: 98.9% accurate, 1.7× speedup

Math

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

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_i_m
    (*
     (+
      (/ 1.0 v)
      (*
       sinTheta_i
       (/
        (- (* (/ 0.5 v) (* sinTheta_i (* sinTheta_O sinTheta_O))) sinTheta_O)
        (* v v))))
     (* cosTheta_O_m (/ (/ 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 * (cosTheta_i_m * (((1.0f / v) + (sinTheta_i * ((((0.5f / v) * (sinTheta_i * (sinTheta_O * sinTheta_O))) - sinTheta_O) / (v * v)))) * (cosTheta_O_m * ((0.5f / v) / sinhf((1.0f / 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_i_m * (((1.0e0 / v) + (sintheta_i * ((((0.5e0 / v) * (sintheta_i * (sintheta_o * sintheta_o))) - sintheta_o) / (v * v)))) * (costheta_o_m * ((0.5e0 / v) / sinh((1.0e0 / 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_i_m * Float32(Float32(Float32(Float32(1.0) / v) + Float32(sinTheta_i * Float32(Float32(Float32(Float32(Float32(0.5) / v) * Float32(sinTheta_i * Float32(sinTheta_O * sinTheta_O))) - sinTheta_O) / Float32(v * v)))) * Float32(cosTheta_O_m * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / 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_i_m * (((single(1.0) / v) + (sinTheta_i * ((((single(0.5) / v) * (sinTheta_i * (sinTheta_O * sinTheta_O))) - sinTheta_O) / (v * v)))) * (cosTheta_O_m * ((single(0.5) / v) / sinh((single(1.0) / v)))))));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\color{blue}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{e^{0 - \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\color{blue}{\left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right) + \frac{1}{v}\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{v} + sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{v}\right), \left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} + -1 \cdot \frac{sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} + \left(\mathsf{neg}\left(\frac{sinTheta\_O}{{v}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} - \frac{sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right), \left(\frac{sinTheta\_O}{{v}^{2}}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

7. Simplified 98.8%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot \left(\left(\frac{1}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{2}}{v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v} - \frac{sinTheta\_O}{v \cdot v}\right)\right) \cdot \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}\right)\right), \color{blue}{cosTheta\_i}\right) \]

9. Applied egg-rr 98.9%

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

10. Final simplification 98.9%

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

Alternative 3: 98.8% accurate, 1.8× speedup

Math

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

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_i_m
     (- (/ cosTheta_O_m (/ v (* sinTheta_i sinTheta_O))) cosTheta_O_m))
    (/ (/ -1.0 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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) * ((-1.0f / 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_i_m * ((costheta_o_m / (v / (sintheta_i * sintheta_o))) - costheta_o_m)) * (((-1.0e0) / 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(cosTheta_i_m * Float32(Float32(cosTheta_O_m / Float32(v / Float32(sinTheta_i * sinTheta_O))) - cosTheta_O_m)) * Float32(Float32(Float32(-1.0) / v) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) * ((single(-1.0) / v) / (sinh((single(1.0) / v)) / (single(0.5) / v)))));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + -1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} - cosTheta\_O \cdot cosTheta\_i\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), 2\right), v\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot sinTheta\_O\right) \cdot sinTheta\_i\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right) \cdot sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   16. neg-lowering-neg.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{neg.f32}\left(v\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \color{blue}{2}\right), v\right)\right) \]

5. Simplified 98.4%

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

   1. div-inv N/A

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

   2. times-frac N/A

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

   3. distribute-frac-neg2 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_i}{v} - cosTheta\_O \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right), \color{blue}{\left(\frac{\frac{1}{v}}{\mathsf{neg}\left(v\right)}\right)}\right) \]

7. Applied egg-rr 97.9%

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

   1. frac-times N/A

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

   2. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}} - cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)}\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right) - cosTheta\_i \cdot cosTheta\_O\right), \left(\frac{\frac{\color{blue}{-1}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

   5. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}} - cosTheta\_O\right)\right), \left(\frac{\color{blue}{\frac{-1}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}} - cosTheta\_O\right)\right), \left(\frac{\color{blue}{\frac{-1}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\left(cosTheta\_O \cdot \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{\color{blue}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\left(cosTheta\_O \cdot \frac{1}{\frac{\frac{v}{sinTheta\_O}}{sinTheta\_i}}\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

   9. un-div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\left(\frac{cosTheta\_O}{\frac{\frac{v}{sinTheta\_O}}{sinTheta\_i}}\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{\frac{v}{sinTheta\_O}}{sinTheta\_i}\right)\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

   11. associate-/l/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{v}{sinTheta\_i \cdot sinTheta\_O}\right)\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \left(sinTheta\_i \cdot sinTheta\_O\right)\right)\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{v \cdot \color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}}\right)\right) \]

9. Applied egg-rr 98.7%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

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 (+ (/ (* sinTheta_i sinTheta_O) v) 1.0))
    (* (/ (/ 0.5 v) (sinh (/ 1.0 v))) (/ 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 * ((cosTheta_O_m / (((sinTheta_i * sinTheta_O) / v) + 1.0f)) * (((0.5f / v) / sinhf((1.0f / v))) * (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 * ((costheta_o_m / (((sintheta_i * sintheta_o) / v) + 1.0e0)) * (((0.5e0 / v) / sinh((1.0e0 / v))) * (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(cosTheta_O_m / Float32(Float32(Float32(sinTheta_i * sinTheta_O) / v) + Float32(1.0))) * Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))) * Float32(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 * ((cosTheta_O_m / (((sinTheta_i * sinTheta_O) / v) + single(1.0))) * (((single(0.5) / v) / sinh((single(1.0) / v))) * (cosTheta_i_m / v))));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{cosTheta\_O}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \color{blue}{\left(1 + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v} + 1\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{+.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), 1\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right), 1\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f32 98.6%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right), 1\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 98.7% accurate, 1.9× speedup

Math

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

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 (/ (/ 1.0 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 * ((1.0f / 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 * ((1.0e0 / 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(cosTheta_i_m * Float32(Float32(Float32(1.0) / v) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v)))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_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(1.0) / v) / (sinh((single(1.0) / v)) / (single(0.5) / v))))));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\color{blue}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{e^{0 - \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\color{blue}{\left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right) + \frac{1}{v}\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{v} + sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{v}\right), \left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(sinTheta\_i \cdot \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(-1 \cdot \frac{sinTheta\_O}{{v}^{2}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} + -1 \cdot \frac{sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} + \left(\mathsf{neg}\left(\frac{sinTheta\_O}{{v}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}} - \frac{sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2} \cdot sinTheta\_i}{{v}^{3}}\right), \left(\frac{sinTheta\_O}{{v}^{2}}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

7. Simplified 98.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\left(\left(\frac{1}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{2}}{v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v} - \frac{sinTheta\_O}{v \cdot v}\right)\right) \cdot \frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \cdot cosTheta\_i\right), \color{blue}{cosTheta\_O}\right) \]

9. Applied egg-rr 98.9%

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

10. Taylor expanded in v around inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(\frac{1}{v}\right)}, \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), cosTheta\_i\right), cosTheta\_O\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f32 98.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right)\right), cosTheta\_i\right), cosTheta\_O\right) \]

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 6: 98.6% accurate, 1.9× speedup

Math

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

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
   (*
    (* (/ 1.0 v) (/ (/ 0.5 v) (sinh (/ 1.0 v))))
    (* cosTheta_i_m cosTheta_O_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 * (((1.0f / v) * ((0.5f / v) / sinhf((1.0f / v)))) * (cosTheta_i_m * cosTheta_O_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 * (((1.0e0 / v) * ((0.5e0 / v) / sinh((1.0e0 / v)))) * (costheta_i_m * costheta_o_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(Float32(1.0) / v) * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v)))) * Float32(cosTheta_i_m * cosTheta_O_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(1.0) / v) * ((single(0.5) / v) / sinh((single(1.0) / v)))) * (cosTheta_i_m * cosTheta_O_m)));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\color{blue}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{e^{0 - \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\color{blue}{\left(\frac{1}{v}\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 98.6%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 7: 98.6% accurate, 1.9× speedup

Math

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

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 (* (/ (/ 0.5 v) (sinh (/ 1.0 v))) (/ 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 * (cosTheta_O_m * (((0.5f / v) / sinhf((1.0f / v))) * (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 * (costheta_o_m * (((0.5e0 / v) / sinh((1.0e0 / v))) * (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(cosTheta_O_m * Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))) * Float32(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 * (cosTheta_O_m * (((single(0.5) / v) / sinh((single(1.0) / v))) * (cosTheta_i_m / v))));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{cosTheta\_O}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\color{blue}{cosTheta\_O}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 8: 69.7% accurate, 6.3× speedup

Math

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

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_i_m
      (- (/ cosTheta_O_m (/ v (* sinTheta_i sinTheta_O))) cosTheta_O_m))
     (/ v (/ -1.0 v)))
    (/
     0.5
     (/
      (+
       (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
       1.0)
      v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) / (v / (-1.0f / v))) * (0.5f / ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) + 1.0f) / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_m * ((costheta_o_m / (v / (sintheta_i * sintheta_o))) - costheta_o_m)) / (v / ((-1.0e0) / v))) * (0.5e0 / ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) + 1.0e0) / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_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_i_m * Float32(Float32(cosTheta_O_m / Float32(v / Float32(sinTheta_i * sinTheta_O))) - cosTheta_O_m)) / Float32(v / Float32(Float32(-1.0) / v))) * Float32(Float32(0.5) / Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) + Float32(1.0)) / 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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) / (v / (single(-1.0) / v))) * (single(0.5) / ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) + single(1.0)) / v))));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + -1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} - cosTheta\_O \cdot cosTheta\_i\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), 2\right), v\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot sinTheta\_O\right) \cdot sinTheta\_i\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right) \cdot sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   16. neg-lowering-neg.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{neg.f32}\left(v\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \color{blue}{2}\right), v\right)\right) \]

5. Simplified 98.4%

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

   1. div-inv N/A

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

   2. times-frac N/A

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

   3. distribute-frac-neg2 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_i}{v} - cosTheta\_O \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right), \color{blue}{\left(\frac{\frac{1}{v}}{\mathsf{neg}\left(v\right)}\right)}\right) \]

7. Applied egg-rr 97.9%

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

8. Taylor expanded in v around -inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), v\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

10. Simplified 68.2%

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

12. Applied egg-rr 68.3%

    \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}} - cosTheta\_O\right)}{\frac{v}{\frac{-1}{v}}} \cdot \frac{0.5}{\frac{1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}}} \]

13. Final simplification 68.3%

    \[\leadsto \frac{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}} - cosTheta\_O\right)}{\frac{v}{\frac{-1}{v}}} \cdot \frac{0.5}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} + 1}{v}} \]
14. Add Preprocessing

Alternative 9: 69.7% accurate, 6.3× speedup

Math

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

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_i_m
     (- (/ cosTheta_O_m (/ v (* sinTheta_i sinTheta_O))) cosTheta_O_m))
    (/
     (/ 1.0 v)
     (*
      v
      (/
       (*
        2.0
        (-
         -1.0
         (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v 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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) * ((1.0f / v) / (v * ((2.0f * (-1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * 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_i_m * ((costheta_o_m / (v / (sintheta_i * sintheta_o))) - costheta_o_m)) * ((1.0e0 / v) / (v * ((2.0e0 * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * 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_i_m * Float32(Float32(cosTheta_O_m / Float32(v / Float32(sinTheta_i * sinTheta_O))) - cosTheta_O_m)) * Float32(Float32(Float32(1.0) / v) / Float32(v * Float32(Float32(Float32(2.0) * Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * 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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) * ((single(1.0) / v) / (v * ((single(2.0) * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)))) / v)))));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + -1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} - cosTheta\_O \cdot cosTheta\_i\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), 2\right), v\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot sinTheta\_O\right) \cdot sinTheta\_i\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right) \cdot sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   16. neg-lowering-neg.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{neg.f32}\left(v\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \color{blue}{2}\right), v\right)\right) \]

5. Simplified 98.4%

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

   1. div-inv N/A

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

   2. times-frac N/A

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

   3. distribute-frac-neg2 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_i}{v} - cosTheta\_O \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right), \color{blue}{\left(\frac{\frac{1}{v}}{\mathsf{neg}\left(v\right)}\right)}\right) \]

7. Applied egg-rr 97.9%

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

8. Taylor expanded in v around -inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), v\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

10. Simplified 68.2%

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

12. Applied egg-rr 68.3%

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

13. Final simplification 68.3%

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

Alternative 10: 69.7% accurate, 6.7× speedup

Math

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

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_i_m
     (- (/ cosTheta_O_m (/ v (* sinTheta_i sinTheta_O))) cosTheta_O_m))
    (*
     (* v v)
     (/
      (*
       2.0
       (-
        -1.0
        (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v 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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) / ((v * v) * ((2.0f * (-1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * 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_i_m * ((costheta_o_m / (v / (sintheta_i * sintheta_o))) - costheta_o_m)) / ((v * v) * ((2.0e0 * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * 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_i_m * Float32(Float32(cosTheta_O_m / Float32(v / Float32(sinTheta_i * sinTheta_O))) - cosTheta_O_m)) / Float32(Float32(v * v) * Float32(Float32(Float32(2.0) * Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * 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_i_m * ((cosTheta_O_m / (v / (sinTheta_i * sinTheta_O))) - cosTheta_O_m)) / ((v * v) * ((single(2.0) * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)))) / v))));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + -1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} - cosTheta\_O \cdot cosTheta\_i\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), 2\right), v\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot sinTheta\_O\right) \cdot sinTheta\_i\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right) \cdot sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   16. neg-lowering-neg.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{neg.f32}\left(v\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \color{blue}{2}\right), v\right)\right) \]

5. Simplified 98.4%

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

   1. div-inv N/A

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

   2. times-frac N/A

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

   3. distribute-frac-neg2 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_i}{v} - cosTheta\_O \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right), \color{blue}{\left(\frac{\frac{1}{v}}{\mathsf{neg}\left(v\right)}\right)}\right) \]

7. Applied egg-rr 97.9%

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

8. Taylor expanded in v around -inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), v\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

10. Simplified 68.2%

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

12. Applied egg-rr 68.3%

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

13. Final simplification 68.3%

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

Alternative 11: 69.7% accurate, 8.1× speedup

Math

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

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_i_m cosTheta_O_m)
     (*
      2.0
      (/
       (-
        -1.0
        (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v)))
       v)))
    (/ (/ -1.0 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_i_m * cosTheta_O_m) / (2.0f * ((-1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v))) / v))) * ((-1.0f / 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_i_m * costheta_o_m) / (2.0e0 * (((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))) / v))) * (((-1.0e0) / 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(Float32(cosTheta_i_m * cosTheta_O_m) / Float32(Float32(2.0) * Float32(Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v))) / v))) * Float32(Float32(Float32(-1.0) / 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_i_m * cosTheta_O_m) / (single(2.0) * ((single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))) / v))) * ((single(-1.0) / v) / v)));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right) + \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right)}, v\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + -1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} - cosTheta\_O \cdot cosTheta\_i\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, 2\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), 2\right), v\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(\left(cosTheta\_i \cdot sinTheta\_O\right) \cdot sinTheta\_i\right)\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right) \cdot sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, v\right)\right), 2\right), v\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), 2\right), v\right)\right) \]

   16. neg-lowering-neg.f32 98.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), sinTheta\_O\right), sinTheta\_i\right), v\right), \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{neg.f32}\left(v\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \color{blue}{2}\right), v\right)\right) \]

5. Simplified 98.4%

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

   1. div-inv N/A

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

   2. times-frac N/A

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

   3. distribute-frac-neg2 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot sinTheta\_O\right) \cdot sinTheta\_i}{v} - cosTheta\_O \cdot cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right), \color{blue}{\left(\frac{\frac{1}{v}}{\mathsf{neg}\left(v\right)}\right)}\right) \]

7. Applied egg-rr 97.9%

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

8. Taylor expanded in v around -inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), v\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

10. Simplified 68.2%

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

11. Taylor expanded in sinTheta_i around 0

    \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{6}, \mathsf{/.f32}\left(\frac{1}{120}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), -1\right), v\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(-1, \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{6}, \mathsf{/.f32}\left(\frac{1}{120}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), -1\right), v\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{-1}, v\right), v\right)\right) \]

    2. *-lowering-*.f32 68.1%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(-1, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{6}, \mathsf{/.f32}\left(\frac{1}{120}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right), -1\right), v\right)\right), 2\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), v\right)\right) \]

13. Simplified 68.1%

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

14. Final simplification 68.1%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{2 \cdot \frac{-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}}{v}} \cdot \frac{\frac{-1}{v}}{v} \]
15. Add Preprocessing

Alternative 12: 63.4% accurate, 16.9× speedup

Math

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

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_i_m
    (/ (/ cosTheta_O_m (- 2.0 (/ -0.3333333333333333 (* v 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_i_m * ((cosTheta_O_m / (2.0f - (-0.3333333333333333f / (v * 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_i_m * ((costheta_o_m / (2.0e0 - ((-0.3333333333333333e0) / (v * 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_i_m * Float32(Float32(cosTheta_O_m / Float32(Float32(2.0) - Float32(Float32(-0.3333333333333333) / Float32(v * 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_i_m * ((cosTheta_O_m / (single(2.0) - (single(-0.3333333333333333) / (v * v)))) / v)));
end

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 93.2%

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

5. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \color{blue}{\left(2 + -1 \cdot \frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(2 + \left(\mathsf{neg}\left(\frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}\right)\right)\right)\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(2 - \color{blue}{\frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}\right), \color{blue}{v}\right)\right)\right)\right)\right) \]

7. Simplified 60.8%

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

   1. clear-num N/A

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

   2. associate-/l* N/A

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

   3. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{2 - \frac{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -2 + 2 \cdot \frac{\left(\frac{-1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right) + \frac{-1}{6}}{v}}{v}}}{v}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{2 - \frac{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -2 + 2 \cdot \frac{\left(\frac{-1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right) + \frac{-1}{6}}{v}}{v}}\right), \color{blue}{v}\right)\right) \]

9. Applied egg-rr 62.0%

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

10. Taylor expanded in sinTheta_i around 0

    \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{\frac{-1}{3}}{{v}^{2}}\right)}\right)\right), v\right)\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{3}, \left({v}^{2}\right)\right)\right)\right), v\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{3}, \left(v \cdot v\right)\right)\right)\right), v\right)\right) \]

    3. *-lowering-*.f32 62.0%

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right), v\right)\right) \]

12. Simplified 62.0%

    \[\leadsto cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{2 - \color{blue}{\frac{-0.3333333333333333}{v \cdot v}}}}{v} \]
13. Add Preprocessing

Alternative 13: 63.4% accurate, 16.9× speedup

Math

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

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_i_m
    (/ (/ cosTheta_O_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_i_m * ((cosTheta_O_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_i_m * ((costheta_o_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_i_m * Float32(Float32(cosTheta_O_m / 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_i_m * ((cosTheta_O_m / v) / (single(2.0) + (single(0.3333333333333333) / (v * v))))));
end

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 93.2%

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

5. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \color{blue}{\left(2 + -1 \cdot \frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(2 + \left(\mathsf{neg}\left(\frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}\right)\right)\right)\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(2 - \color{blue}{\frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}}{v}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(\left(-2 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + 2 \cdot \frac{\frac{-1}{2} \cdot \left({sinTheta\_O}^{2} \cdot {sinTheta\_i}^{2}\right) - \frac{1}{6}}{v}\right), \color{blue}{v}\right)\right)\right)\right)\right) \]

7. Simplified 60.8%

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

   1. clear-num N/A

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

   2. associate-/l* N/A

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

   3. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{2 - \frac{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -2 + 2 \cdot \frac{\left(\frac{-1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right) + \frac{-1}{6}}{v}}{v}}}{v}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{2 - \frac{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -2 + 2 \cdot \frac{\left(\frac{-1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right) \cdot \left(sinTheta\_i \cdot sinTheta\_i\right) + \frac{-1}{6}}{v}}{v}}\right), \color{blue}{v}\right)\right) \]

9. Applied egg-rr 62.0%

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

10. Taylor expanded in sinTheta_i around 0

    \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{cosTheta\_O}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)}\right) \]
11. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \left(\frac{\frac{cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{v}\right), \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \left(\color{blue}{2} + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{+.f32}\left(2, \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right)\right) \]

    5. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{\color{blue}{v}}^{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right) \]

    9. *-lowering-*.f32 62.0%

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right)\right) \]

12. Simplified 62.0%

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

Alternative 14: 58.0% accurate, 24.4× speedup

Math

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

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 (/ 1.0 (/ (* v 2.0) (* cosTheta_i_m cosTheta_O_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 * (1.0f / ((v * 2.0f) / (cosTheta_i_m * cosTheta_O_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 * (1.0e0 / ((v * 2.0e0) / (costheta_i_m * costheta_o_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(1.0) / Float32(Float32(v * Float32(2.0)) / Float32(cosTheta_i_m * cosTheta_O_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(1.0) / ((v * single(2.0)) / (cosTheta_i_m * cosTheta_O_m))));
end

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 93.2%

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

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{2 \cdot v}{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(2 \cdot v\right), \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(2, v\right), \left(\color{blue}{cosTheta\_O} \cdot cosTheta\_i\right)\right)\right) \]

   4. *-lowering-*.f32 56.5%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(2, v\right), \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{cosTheta\_i}\right)\right)\right) \]

7. Simplified 56.5%

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

8. Final simplification 56.5%

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

Alternative 15: 57.6% accurate, 31.4× speedup

Math

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

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_i_m cosTheta_O_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_i_m * cosTheta_O_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_i_m * costheta_o_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_i_m * cosTheta_O_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_i_m * cosTheta_O_m)) / v));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\color{blue}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{e^{0 - \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{2}\right), v\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \frac{1}{2}\right), v\right) \]

   5. *-lowering-*.f32 56.1%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \frac{1}{2}\right), v\right) \]

7. Simplified 56.1%

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

8. Final simplification 56.1%

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

Alternative 16: 57.6% accurate, 31.4× speedup

Math

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

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_i_m cosTheta_O_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_i_m * cosTheta_O_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_i_m * costheta_o_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_i_m * cosTheta_O_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_i_m * cosTheta_O_m)));
end

Derivation

1. Initial program 98.4%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. exp-neg N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

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

       \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\color{blue}{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{*.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot v}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{e^{0 - \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{v} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right) \]

7. Simplified 56.1%

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

8. Final simplification 56.1%

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

Alternative 17: 57.6% accurate, 31.4× speedup

Math

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

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 (* cosTheta_i_m (* (/ 0.5 v) cosTheta_O_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_i_m * ((0.5f / v) * cosTheta_O_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 * (costheta_i_m * ((0.5e0 / v) * costheta_o_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(cosTheta_i_m * Float32(Float32(Float32(0.5) / v) * cosTheta_O_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 * (cosTheta_i_m * ((single(0.5) / v) * cosTheta_O_m)));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\left(\frac{cosTheta\_i}{v}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   6. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \frac{1}{2}\right)\right) \]

5. Simplified 56.1%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_i}{v}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{\frac{1}{2}}{\color{blue}{\frac{v}{cosTheta\_i}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{v}{cosTheta\_i}\right)}\right)\right) \]

   5. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{cosTheta\_i}\right)\right)\right) \]

7. Applied egg-rr 56.1%

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

   1. associate-/r/ N/A

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

   2. associate-*r* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot \frac{\frac{1}{2}}{v}\right), \color{blue}{cosTheta\_i}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{\frac{1}{2}}{v}\right)\right), cosTheta\_i\right) \]

   5. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), cosTheta\_i\right) \]

9. Applied egg-rr 56.1%

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

10. Final simplification 56.1%

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

Alternative 18: 57.6% accurate, 31.4× speedup

Math

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

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 (* cosTheta_O_m (* (/ 0.5 v) 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 * ((0.5f / v) * 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 * (costheta_o_m * ((0.5e0 / v) * 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(cosTheta_O_m * Float32(Float32(Float32(0.5) / v) * 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 * (cosTheta_O_m * ((single(0.5) / v) * cosTheta_i_m)));
end

Derivation

1. Initial program 98.4%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\left(\frac{cosTheta\_i}{v}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   6. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \frac{1}{2}\right)\right) \]

5. Simplified 56.1%

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

   1. associate-*l/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i \cdot \frac{1}{2}}{\color{blue}{v}}\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i \cdot \color{blue}{\frac{\frac{1}{2}}{v}}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{\frac{1}{2}}{v} \cdot \color{blue}{cosTheta\_i}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \color{blue}{cosTheta\_i}\right)\right) \]

   5. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), cosTheta\_i\right)\right) \]

7. Applied egg-rr 56.1%

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

Alternative 19: 57.6% accurate, 31.4× speedup

Math

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

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 (* 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(0.5) * Float32(cosTheta_O_m * Float32(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 98.4%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\left(\frac{cosTheta\_i}{v}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   6. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \frac{1}{2}\right)\right) \]

5. Simplified 56.1%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_i}{v}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{\frac{1}{2}}{\color{blue}{\frac{v}{cosTheta\_i}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{v}{cosTheta\_i}\right)}\right)\right) \]

   5. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{cosTheta\_i}\right)\right)\right) \]

7. Applied egg-rr 56.1%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(\frac{cosTheta\_i}{v}\right), \color{blue}{cosTheta\_O}\right)\right) \]

   7. /-lowering-/.f32 56.1%

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), cosTheta\_O\right)\right) \]

9. Applied egg-rr 56.1%

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

10. Final simplification 56.1%

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

Reproduce

herbie shell --seed 2024158 
(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)))