HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 21.5s

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{-1}{v}}{\frac{-1 - sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + 0.5 \cdot \frac{\frac{sinTheta\_O \cdot sinTheta\_O}{v}}{v}\right)\right)}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (/
   (/ -1.0 v)
   (/
    (-
     -1.0
     (*
      sinTheta_i
      (+
       (/ sinTheta_O v)
       (*
        sinTheta_i
        (+
         (/
          (*
           0.16666666666666666
           (* sinTheta_i (* sinTheta_O (* sinTheta_O sinTheta_O))))
          (* v (* v v)))
         (* 0.5 (/ (/ (* sinTheta_O sinTheta_O) v) v)))))))
    (/ (/ 0.5 v) (sinh (/ 1.0 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * ((-1.0f / v) / ((-1.0f - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((0.16666666666666666f * (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O)))) / (v * (v * v))) + (0.5f * (((sinTheta_O * sinTheta_O) / v) / v))))))) / ((0.5f / v) / sinhf((1.0f / 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 = (costheta_i * costheta_o) * (((-1.0e0) / v) / (((-1.0e0) - (sintheta_i * ((sintheta_o / v) + (sintheta_i * (((0.16666666666666666e0 * (sintheta_i * (sintheta_o * (sintheta_o * sintheta_o)))) / (v * (v * v))) + (0.5e0 * (((sintheta_o * sintheta_o) / v) / v))))))) / ((0.5e0 / v) / sinh((1.0e0 / v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(Float32(-1.0) / v) / Float32(Float32(Float32(-1.0) - Float32(sinTheta_i * Float32(Float32(sinTheta_O / v) + Float32(sinTheta_i * Float32(Float32(Float32(Float32(0.16666666666666666) * Float32(sinTheta_i * Float32(sinTheta_O * Float32(sinTheta_O * sinTheta_O)))) / Float32(v * Float32(v * v))) + Float32(Float32(0.5) * Float32(Float32(Float32(sinTheta_O * sinTheta_O) / v) / v))))))) / Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * ((single(-1.0) / v) / ((single(-1.0) - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((single(0.16666666666666666) * (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O)))) / (v * (v * v))) + (single(0.5) * (((sinTheta_O * sinTheta_O) / v) / v))))))) / ((single(0.5) / v) / sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. div-inv N/A

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

   2. associate-/l* N/A

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

   3. remove-double-neg N/A

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

   4. div-inv N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(0 - cosTheta\_i \cdot cosTheta\_O\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \left(cosTheta\_i \cdot cosTheta\_O\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right), \color{blue}{\left(\frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \color{blue}{\left(1 + sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)\right)}\right)\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \color{blue}{\left(sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)}\right)\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{sinTheta\_O}{v} + \color{blue}{sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\left(\frac{sinTheta\_O}{v}\right), \color{blue}{\left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \left(\color{blue}{sinTheta\_i} \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 98.9%

   \[\leadsto \left(0 - cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}\right) \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\frac{-1}{v} \cdot \frac{1}{\color{blue}{\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}}}\right)\right) \]

    2. un-div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\color{blue}{\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\left(\frac{-1}{v}\right), \color{blue}{\left(\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \left(\frac{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\left(1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}\right)}\right)\right)\right) \]

11. Applied egg-rr 98.9%

    \[\leadsto \left(0 - cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{\frac{-1}{v}}{\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + 0.5 \cdot \frac{\frac{sinTheta\_O \cdot sinTheta\_O}{v}}{v}\right)\right)}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}}} \]

12. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}{-1 - sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot 0.5}{v \cdot v}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (*
   (/ -1.0 v)
   (/
    (/ (/ 0.5 v) (sinh (/ 1.0 v)))
    (-
     -1.0
     (*
      sinTheta_i
      (+
       (/ sinTheta_O v)
       (*
        sinTheta_i
        (+
         (/
          (*
           0.16666666666666666
           (* sinTheta_i (* sinTheta_O (* sinTheta_O sinTheta_O))))
          (* v (* v v)))
         (/ (* (* sinTheta_O sinTheta_O) 0.5) (* v v)))))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * ((-1.0f / v) * (((0.5f / v) / sinhf((1.0f / v))) / (-1.0f - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((0.16666666666666666f * (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O)))) / (v * (v * v))) + (((sinTheta_O * sinTheta_O) * 0.5f) / (v * 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 = (costheta_i * costheta_o) * (((-1.0e0) / v) * (((0.5e0 / v) / sinh((1.0e0 / v))) / ((-1.0e0) - (sintheta_i * ((sintheta_o / v) + (sintheta_i * (((0.16666666666666666e0 * (sintheta_i * (sintheta_o * (sintheta_o * sintheta_o)))) / (v * (v * v))) + (((sintheta_o * sintheta_o) * 0.5e0) / (v * v)))))))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(Float32(-1.0) / v) * Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))) / Float32(Float32(-1.0) - Float32(sinTheta_i * Float32(Float32(sinTheta_O / v) + Float32(sinTheta_i * Float32(Float32(Float32(Float32(0.16666666666666666) * Float32(sinTheta_i * Float32(sinTheta_O * Float32(sinTheta_O * sinTheta_O)))) / Float32(v * Float32(v * v))) + Float32(Float32(Float32(sinTheta_O * sinTheta_O) * Float32(0.5)) / Float32(v * v))))))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * ((single(-1.0) / v) * (((single(0.5) / v) / sinh((single(1.0) / v))) / (single(-1.0) - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((single(0.16666666666666666) * (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O)))) / (v * (v * v))) + (((sinTheta_O * sinTheta_O) * single(0.5)) / (v * v)))))))));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. div-inv N/A

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

   2. associate-/l* N/A

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

   3. remove-double-neg N/A

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

   4. div-inv N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(0 - cosTheta\_i \cdot cosTheta\_O\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \left(cosTheta\_i \cdot cosTheta\_O\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right), \color{blue}{\left(\frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \color{blue}{\left(1 + sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)\right)}\right)\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \color{blue}{\left(sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)}\right)\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{sinTheta\_O}{v} + \color{blue}{sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\left(\frac{sinTheta\_O}{v}\right), \color{blue}{\left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \left(\color{blue}{sinTheta\_i} \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 98.9%

   \[\leadsto \left(0 - cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}\right) \]

10. Final simplification 98.9%

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

Alternative 3: 98.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{-1}{v}}{\frac{-1 - sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \frac{\left(sinTheta\_O \cdot sinTheta\_O\right) \cdot 0.5}{v \cdot v}\right)}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (/
   (/ -1.0 v)
   (/
    (-
     -1.0
     (*
      sinTheta_i
      (+
       (/ sinTheta_O v)
       (* sinTheta_i (/ (* (* sinTheta_O sinTheta_O) 0.5) (* v v))))))
    (/ (/ 0.5 v) (sinh (/ 1.0 v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * ((-1.0f / v) / ((-1.0f - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((sinTheta_O * sinTheta_O) * 0.5f) / (v * v)))))) / ((0.5f / v) / sinhf((1.0f / 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 = (costheta_i * costheta_o) * (((-1.0e0) / v) / (((-1.0e0) - (sintheta_i * ((sintheta_o / v) + (sintheta_i * (((sintheta_o * sintheta_o) * 0.5e0) / (v * v)))))) / ((0.5e0 / v) / sinh((1.0e0 / v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(Float32(-1.0) / v) / Float32(Float32(Float32(-1.0) - Float32(sinTheta_i * Float32(Float32(sinTheta_O / v) + Float32(sinTheta_i * Float32(Float32(Float32(sinTheta_O * sinTheta_O) * Float32(0.5)) / Float32(v * v)))))) / Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * ((single(-1.0) / v) / ((single(-1.0) - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((sinTheta_O * sinTheta_O) * single(0.5)) / (v * v)))))) / ((single(0.5) / v) / sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. div-inv N/A

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

   2. associate-/l* N/A

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

   3. remove-double-neg N/A

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

   4. div-inv N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(0 - cosTheta\_i \cdot cosTheta\_O\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \left(cosTheta\_i \cdot cosTheta\_O\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right), \color{blue}{\left(\frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \color{blue}{\left(1 + sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)\right)}\right)\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \color{blue}{\left(sinTheta\_i \cdot \left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right) + \frac{sinTheta\_O}{v}\right)}\right)\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{sinTheta\_O}{v} + \color{blue}{sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\left(\frac{sinTheta\_O}{v}\right), \color{blue}{\left(sinTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \left(\color{blue}{sinTheta\_i} \cdot \left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\left(\frac{1}{6} \cdot \frac{{sinTheta\_O}^{3} \cdot sinTheta\_i}{{v}^{3}}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 98.9%

   \[\leadsto \left(0 - cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}\right) \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\frac{-1}{v} \cdot \frac{1}{\color{blue}{\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}}}\right)\right) \]

    2. un-div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\frac{\frac{-1}{v}}{\color{blue}{\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\left(\frac{-1}{v}\right), \color{blue}{\left(\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \left(\frac{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\left(1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{\frac{1}{6} \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + \frac{\frac{1}{2} \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}\right)}\right)\right)\right) \]

11. Applied egg-rr 98.9%

    \[\leadsto \left(0 - cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{\frac{-1}{v}}{\frac{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)} + 0.5 \cdot \frac{\frac{sinTheta\_O \cdot sinTheta\_O}{v}}{v}\right)\right)}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}}} \]

12. Taylor expanded in sinTheta_i around 0

    \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \color{blue}{\left(\frac{1}{2} \cdot \frac{{sinTheta\_O}^{2}}{{v}^{2}}\right)}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \left(\frac{\frac{1}{2} \cdot {sinTheta\_O}^{2}}{{v}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot {sinTheta\_O}^{2}\right), \left({v}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left({sinTheta\_O}^{2}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(sinTheta\_O \cdot sinTheta\_O\right)\right), \left({v}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \left({v}^{2}\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \left(v \cdot v\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f32 98.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right)\right) \]

14. Simplified 98.9%

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

15. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_O + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{\frac{v}{-0.5}}\right)}{v} - cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{\frac{-0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/
   (-
    (/
     (*
      (* cosTheta_i cosTheta_O)
      (+
       (* sinTheta_i sinTheta_O)
       (/ (* sinTheta_O (* sinTheta_i (* sinTheta_i sinTheta_O))) (/ v -0.5))))
     v)
    (* cosTheta_i cosTheta_O))
   v)
  (/ (/ -0.5 v) (sinh (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((((cosTheta_i * cosTheta_O) * ((sinTheta_i * sinTheta_O) + ((sinTheta_O * (sinTheta_i * (sinTheta_i * sinTheta_O))) / (v / -0.5f)))) / v) - (cosTheta_i * cosTheta_O)) / v) * ((-0.5f / v) / sinhf((1.0f / 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 = (((((costheta_i * costheta_o) * ((sintheta_i * sintheta_o) + ((sintheta_o * (sintheta_i * (sintheta_i * sintheta_o))) / (v / (-0.5e0))))) / v) - (costheta_i * costheta_o)) / v) * (((-0.5e0) / v) / sinh((1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(sinTheta_i * sinTheta_O) + Float32(Float32(sinTheta_O * Float32(sinTheta_i * Float32(sinTheta_i * sinTheta_O))) / Float32(v / Float32(-0.5))))) / v) - Float32(cosTheta_i * cosTheta_O)) / v) * Float32(Float32(Float32(-0.5) / v) / sinh(Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((((cosTheta_i * cosTheta_O) * ((sinTheta_i * sinTheta_O) + ((sinTheta_O * (sinTheta_i * (sinTheta_i * sinTheta_O))) / (v / single(-0.5))))) / v) - (cosTheta_i * cosTheta_O)) / v) * ((single(-0.5) / v) / sinh((single(1.0) / v)));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 98.5%

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

9. Applied egg-rr 98.7%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{1}{v}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_i cosTheta_O) (* (/ (/ 0.5 v) (sinh (/ 1.0 v))) (/ 1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (((0.5f / v) / sinhf((1.0f / v))) * (1.0f / 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 = (costheta_i * costheta_o) * (((0.5e0 / v) / sinh((1.0e0 / v))) * (1.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * (((single(0.5) / v) / sinh((single(1.0) / v))) * (single(1.0) / v));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. div-inv N/A

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

   2. associate-/l* N/A

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

   3. remove-double-neg N/A

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

   4. div-inv N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(0 - cosTheta\_i \cdot cosTheta\_O\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \left(cosTheta\_i \cdot cosTheta\_O\right)\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot \frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right), \color{blue}{\left(\frac{\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right), \color{blue}{1}\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (/ 0.5 v) (sinh (/ 1.0 v))) (/ (* cosTheta_i cosTheta_O) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((0.5f / v) / sinhf((1.0f / v))) * ((cosTheta_i * cosTheta_O) / 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 = ((0.5e0 / v) / sinh((1.0e0 / v))) * ((costheta_i * costheta_o) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(0.5) / v) / sinh((single(1.0) / v))) * ((cosTheta_i * cosTheta_O) / v);
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. sinh-undef N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.3%

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

    1. frac-times N/A

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

    2. div-inv N/A

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

    3. associate-*l* N/A

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

    4. metadata-eval N/A

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

    5. associate-/r/ N/A

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

    6. div-inv N/A

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

    7. clear-num N/A

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

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

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

    9. associate-*r/ N/A

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

    10. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

    16. /-lowering-/.f32 98.6%

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

11. Applied egg-rr 98.6%

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

12. Final simplification 98.6%

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

Alternative 7: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_i}{\frac{v}{cosTheta\_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (/ 0.5 v) (sinh (/ 1.0 v))) (/ cosTheta_i (/ v cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((0.5f / v) / sinhf((1.0f / v))) * (cosTheta_i / (v / cosTheta_O));
}

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 = ((0.5e0 / v) / sinh((1.0e0 / v))) * (costheta_i / (v / costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v))) * Float32(cosTheta_i / Float32(v / cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(0.5) / v) / sinh((single(1.0) / v))) * (cosTheta_i / (v / cosTheta_O));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. sinh-undef N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.3%

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

    1. *-commutative N/A

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

    2. associate-/l/ N/A

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

    3. associate-/r* N/A

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

    4. frac-times N/A

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

    5. associate-/l* N/A

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

    6. metadata-eval N/A

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

    7. div-inv N/A

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

    8. associate-*r/ N/A

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

    9. clear-num N/A

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

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

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

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

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

    12. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

    18. /-lowering-/.f32 93.5%

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

11. Applied egg-rr 93.5%

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

    1. clear-num N/A

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

    2. div-inv N/A

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

    3. associate-/l* N/A

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

    4. times-frac N/A

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

    5. frac-2neg N/A

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

    6. un-div-inv N/A

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

    7. metadata-eval N/A

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

    8. frac-2neg N/A

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

    9. un-div-inv N/A

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

    10. clear-num N/A

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

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

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

    12. frac-2neg N/A

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

    13. metadata-eval N/A

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

    14. un-div-inv N/A

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

    15. frac-2neg N/A

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

    16. associate-*r/ N/A

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

    17. clear-num N/A

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

    18. un-div-inv N/A

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

    19. /-lowering-/.f32 N/A

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

    20. /-lowering-/.f32 N/A

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

    21. *-commutative N/A

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

13. Applied egg-rr 98.2%

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

14. Final simplification 98.2%

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

Alternative 8: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (/ (/ cosTheta_i v) (/ (sinh (/ 1.0 v)) (/ 0.5 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i / v) / (sinhf((1.0f / v)) / (0.5f / 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 = costheta_o * ((costheta_i / v) / (sinh((1.0e0 / v)) / (0.5e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(cosTheta_i / v) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) / (sinh((single(1.0) / v)) / (single(0.5) / v)));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. sinh-undef N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.3%

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

    1. frac-times N/A

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

    2. associate-/l* N/A

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

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

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

    4. associate-*l* N/A

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

    5. metadata-eval N/A

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

    6. associate-/r/ N/A

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

    7. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

    13. /-lowering-/.f32 98.5%

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

11. Applied egg-rr 98.5%

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

Alternative 9: 70.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_O + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{\frac{v}{-0.5}}\right)}{v} - cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{-1}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (-
   (/
    (*
     (* cosTheta_i cosTheta_O)
     (+
      (* sinTheta_i sinTheta_O)
      (/ (* sinTheta_O (* sinTheta_i (* sinTheta_i sinTheta_O))) (/ v -0.5))))
    v)
   (* cosTheta_i cosTheta_O))
  (/
   -1.0
   (*
    v
    (-
     (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
     -2.0)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((((cosTheta_i * cosTheta_O) * ((sinTheta_i * sinTheta_O) + ((sinTheta_O * (sinTheta_i * (sinTheta_i * sinTheta_O))) / (v / -0.5f)))) / v) - (cosTheta_i * cosTheta_O)) * (-1.0f / (v * (((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (v * v)) - -2.0f)));
}

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 = ((((costheta_i * costheta_o) * ((sintheta_i * sintheta_o) + ((sintheta_o * (sintheta_i * (sintheta_i * sintheta_o))) / (v / (-0.5e0))))) / v) - (costheta_i * costheta_o)) * ((-1.0e0) / (v * (((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(sinTheta_i * sinTheta_O) + Float32(Float32(sinTheta_O * Float32(sinTheta_i * Float32(sinTheta_i * sinTheta_O))) / Float32(v / Float32(-0.5))))) / v) - Float32(cosTheta_i * cosTheta_O)) * Float32(Float32(-1.0) / Float32(v * Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((((cosTheta_i * cosTheta_O) * ((sinTheta_i * sinTheta_O) + ((sinTheta_O * (sinTheta_i * (sinTheta_i * sinTheta_O))) / (v / single(-0.5))))) / v) - (cosTheta_i * cosTheta_O)) * (single(-1.0) / (v * (((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v)) - single(-2.0))));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 98.5%

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

9. Applied egg-rr 98.4%

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

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), \mathsf{/.f32}\left(v, \frac{-1}{2}\right)\right)\right)\right), v\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right)\right), \mathsf{/.f32}\left(1, \color{blue}{\left(-1 \cdot \left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)}\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), \mathsf{/.f32}\left(v, \frac{-1}{2}\right)\right)\right)\right), v\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right)\right), \mathsf{/.f32}\left(1, \left(\mathsf{neg}\left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), \mathsf{/.f32}\left(v, \frac{-1}{2}\right)\right)\right)\right), v\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right)\right), \mathsf{/.f32}\left(1, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right) \cdot v\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), \mathsf{/.f32}\left(v, \frac{-1}{2}\right)\right)\right)\right), v\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right)\right), \mathsf{/.f32}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right) \cdot \color{blue}{\left(\mathsf{neg}\left(v\right)\right)}\right)\right)\right) \]

    4. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), \mathsf{/.f32}\left(v, \frac{-1}{2}\right)\right)\right)\right), v\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right)\right), \mathsf{/.f32}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right) \cdot \left(-1 \cdot \color{blue}{v}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right), \mathsf{/.f32}\left(v, \frac{-1}{2}\right)\right)\right)\right), v\right), \mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right)\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right), \color{blue}{\left(-1 \cdot v\right)}\right)\right)\right) \]

12. Simplified 67.3%

    \[\leadsto \left(-\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_O + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{\frac{v}{-0.5}}\right)}{v} - cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{1}{\color{blue}{\left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{-v \cdot v} + -2\right) \cdot \left(-v\right)}} \]

13. Final simplification 67.3%

    \[\leadsto \left(\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_O + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{\frac{v}{-0.5}}\right)}{v} - cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{-1}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)} \]
14. Add Preprocessing

Alternative 10: 64.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right) \cdot \frac{1}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (-
   (* cosTheta_i cosTheta_O)
   (/ (* cosTheta_O (* cosTheta_i (* sinTheta_i sinTheta_O))) v))
  (/ 1.0 (* v (+ 2.0 (/ 0.3333333333333333 (* v v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) - ((cosTheta_O * (cosTheta_i * (sinTheta_i * sinTheta_O))) / v)) * (1.0f / (v * (2.0f + (0.3333333333333333f / (v * 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 = ((costheta_i * costheta_o) - ((costheta_o * (costheta_i * (sintheta_i * sintheta_o))) / v)) * (1.0e0 / (v * (2.0e0 + (0.3333333333333333e0 / (v * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(Float32(cosTheta_O * Float32(cosTheta_i * Float32(sinTheta_i * sinTheta_O))) / v)) * Float32(Float32(1.0) / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) - ((cosTheta_O * (cosTheta_i * (sinTheta_i * sinTheta_O))) / v)) * (single(1.0) / (v * (single(2.0) + (single(0.3333333333333333) / (v * v)))));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 98.5%

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

9. Applied egg-rr 98.4%

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

10. Taylor expanded in v around inf

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

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

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

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

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

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

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

    6. unpow2 N/A

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

    7. *-lowering-*.f32 61.0%

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

12. Simplified 61.0%

    \[\leadsto \left(-\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_O + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{\frac{v}{-0.5}}\right)}{v} - cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{1}{\color{blue}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}} \]

13. Taylor expanded in sinTheta_i around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. sub-neg N/A

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

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

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

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

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

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

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

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

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

    8. *-commutative N/A

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

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

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

    10. *-lowering-*.f32 61.0%

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

15. Simplified 61.0%

    \[\leadsto \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot cosTheta\_i\right)}{v}\right)} \cdot \frac{1}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \]

16. Final simplification 61.0%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{v}\right) \cdot \frac{1}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \]
17. Add Preprocessing

Alternative 11: 70.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{\frac{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2}{v}} \cdot \frac{\frac{cosTheta\_i}{v}}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/
   cosTheta_O
   (/
    (-
     (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
     -2.0)
    v))
  (/ (/ cosTheta_i v) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / ((((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (v * v)) - -2.0f) / v)) * ((cosTheta_i / v) / 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 = (costheta_o / ((((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0)) / v)) * ((costheta_i / v) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / Float32(Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0)) / v)) * Float32(Float32(cosTheta_i / v) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / ((((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v)) - single(-2.0)) / v)) * ((cosTheta_i / v) / v);
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. sinh-undef N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.3%

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

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{v}\right)}\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), v\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{v}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right), v\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{\mathsf{neg}\left(v\right)}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right), v\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{-1 \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), v\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right), \left(-1 \cdot v\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right), v\right)\right) \]

12. Simplified 67.2%

    \[\leadsto \frac{cosTheta\_O}{\color{blue}{\frac{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{-v \cdot v} + -2}{-v}}} \cdot \frac{\frac{cosTheta\_i}{v}}{v} \]

13. Final simplification 67.2%

    \[\leadsto \frac{cosTheta\_O}{\frac{\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2}{v}} \cdot \frac{\frac{cosTheta\_i}{v}}{v} \]
14. Add Preprocessing

Alternative 12: 64.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i}{v}}{v} \cdot \frac{cosTheta\_O}{\frac{2 + \frac{0.3333333333333333}{v \cdot v}}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (/ cosTheta_i v) v)
  (/ cosTheta_O (/ (+ 2.0 (/ 0.3333333333333333 (* v v))) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i / v) / v) * (cosTheta_O / ((2.0f + (0.3333333333333333f / (v * v))) / 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 = ((costheta_i / v) / v) * (costheta_o / ((2.0e0 + (0.3333333333333333e0 / (v * v))) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i / v) / v) * Float32(cosTheta_O / Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / v) / v) * (cosTheta_O / ((single(2.0) + (single(0.3333333333333333) / (v * v))) / v));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. sinh-undef N/A

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

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

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.3%

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

10. Taylor expanded in v around inf

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

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

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

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

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

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

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

    6. unpow2 N/A

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

    7. *-lowering-*.f32 61.0%

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

12. Simplified 61.0%

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

13. Final simplification 61.0%

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

Alternative 13: 64.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{cosTheta\_i}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (/ cosTheta_i (* v (+ 2.0 (/ 0.3333333333333333 (* v v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (cosTheta_i / (v * (2.0f + (0.3333333333333333f / (v * 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 = costheta_o * (costheta_i / (v * (2.0e0 + (0.3333333333333333e0 / (v * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(cosTheta_i / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (cosTheta_i / (v * (single(2.0) + (single(0.3333333333333333) / (v * v)))));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 98.5%

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

9. Applied egg-rr 98.4%

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

10. Taylor expanded in v around inf

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

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

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

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

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

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

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

    6. unpow2 N/A

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

    7. *-lowering-*.f32 61.0%

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

12. Simplified 61.0%

    \[\leadsto \left(-\left(\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_O + \frac{sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)\right)}{\frac{v}{-0.5}}\right)}{v} - cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{1}{\color{blue}{v \cdot \left(2 + \frac{0.3333333333333333}{v \cdot v}\right)}} \]

13. Taylor expanded in sinTheta_i around 0

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

    1. associate-/l* N/A

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

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

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

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

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

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

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

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

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f32 61.0%

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

15. Simplified 61.0%

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

Alternative 14: 58.9% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{v}{cosTheta\_i}}{cosTheta\_O \cdot 0.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ (/ v cosTheta_i) (* cosTheta_O 0.5))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / ((v / cosTheta_i) / (cosTheta_O * 0.5f));
}

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 = 1.0e0 / ((v / costheta_i) / (costheta_o * 0.5e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(Float32(v / cosTheta_i) / Float32(cosTheta_O * Float32(0.5))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((v / cosTheta_i) / (cosTheta_O * single(0.5)));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

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 55.4%

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

7. Simplified 55.4%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. frac-2neg N/A

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

   4. *-commutative N/A

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

   5. un-div-inv N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

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

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-out N/A

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

   11. frac-2neg N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

   14. associate-/l* N/A

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

   15. distribute-neg-frac2 N/A

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

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

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

   17. remove-double-neg N/A

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

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

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

   19. /-lowering-/.f32 55.3%

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

9. Applied egg-rr 55.3%

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

    1. associate-*r* N/A

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

    2. clear-num N/A

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

    3. un-div-inv N/A

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

    4. clear-num N/A

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

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

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

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

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

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

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

    8. *-lowering-*.f32 55.5%

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

11. Applied egg-rr 55.5%

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

12. Final simplification 55.5%

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

Alternative 15: 58.9% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ v (* (* cosTheta_i cosTheta_O) 0.5))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / ((cosTheta_i * cosTheta_O) * 0.5f));
}

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 = 1.0e0 / (v / ((costheta_i * costheta_o) * 0.5e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(v / Float32(Float32(cosTheta_i * cosTheta_O) * Float32(0.5))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / ((cosTheta_i * cosTheta_O) * single(0.5)));
end

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

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 55.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \frac{1}{2}\right), v\right) \]

7. Simplified 55.4%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{\mathsf{neg}\left(v\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v\right)} \]

   5. un-div-inv N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{v}\right)}\right) \]

   7. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{-1}{\color{blue}{v}}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{-1}{v}\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right) \cdot \frac{-1}{v}\right)\right) \]

   10. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{-1}{v}\right)\right)\right) \]

   11. frac-2neg N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   13. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)\right)\right)\right)\right) \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)\right)\right)\right)}\right)\right) \]

   17. remove-double-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v}}\right)\right) \]

   18. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right)\right) \]

   19. /-lowering-/.f32 55.3%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right)\right) \]

9. Applied egg-rr 55.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot \color{blue}{\frac{1}{2}} \]

    2. associate-*r/ N/A

       \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{1}{2} \]

    3. *-commutative N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{2} \]

    4. associate-*l/ N/A

       \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}}{\color{blue}{v}} \]

    5. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{v}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}}}} \]

    6. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{v}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}}\right)}\right) \]

    7. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}\right)}\right)\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

    9. *-lowering-*.f32 55.5%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \frac{1}{2}\right)\right)\right) \]

11. Applied egg-rr 55.5%

    \[\leadsto \color{blue}{\frac{1}{\frac{v}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}}} \]
12. Add Preprocessing

Alternative 16: 58.9% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (/ v (* cosTheta_i cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / (v / (cosTheta_i * cosTheta_O));
}

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 = 0.5e0 / (v / (costheta_i * costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(v / Float32(cosTheta_i * cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (cosTheta_i * cosTheta_O));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

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 55.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \frac{1}{2}\right), v\right) \]

7. Simplified 55.4%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{\mathsf{neg}\left(v\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v\right)} \]

   5. un-div-inv N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{v}\right)}\right) \]

   7. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{-1}{\color{blue}{v}}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{-1}{v}\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right) \cdot \frac{-1}{v}\right)\right) \]

   10. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{-1}{v}\right)\right)\right) \]

   11. frac-2neg N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   13. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)\right)\right)\right)\right) \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)\right)\right)\right)}\right)\right) \]

   17. remove-double-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v}}\right)\right) \]

   18. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right)\right) \]

   19. /-lowering-/.f32 55.3%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right)\right) \]

9. Applied egg-rr 55.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{v}}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right) \]

    3. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{cosTheta\_O}{v}\right)}\right)\right) \]

    5. /-lowering-/.f32 55.3%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, \color{blue}{v}\right)\right)\right) \]

11. Applied egg-rr 55.3%

    \[\leadsto 0.5 \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v}} \]

    2. clear-num N/A

       \[\leadsto \frac{1}{2} \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

    3. un-div-inv N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{v}{cosTheta\_i \cdot cosTheta\_O}\right)}\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)\right) \]

    6. *-lowering-*.f32 55.5%

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{cosTheta\_O}\right)\right)\right) \]

13. Applied egg-rr 55.5%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]
14. Add Preprocessing

Alternative 17: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{0.5}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_i cosTheta_O) (/ 0.5 v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (0.5f / 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 = (costheta_i * costheta_o) * (0.5e0 / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(0.5) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * (single(0.5) / v);
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

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 55.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \frac{1}{2}\right), v\right) \]

7. Simplified 55.4%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\color{blue}{cosTheta\_O} \cdot cosTheta\_i\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(cosTheta\_i \cdot \color{blue}{cosTheta\_O}\right)\right) \]

   6. *-lowering-*.f32 55.4%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{cosTheta\_O}\right)\right) \]

9. Applied egg-rr 55.4%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)} \]

10. Final simplification 55.4%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{0.5}{v} \]
11. Add Preprocessing

Alternative 18: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \frac{cosTheta\_O}{\frac{v}{0.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ cosTheta_O (/ v 0.5))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * (cosTheta_O / (v / 0.5f));
}

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 = costheta_i * (costheta_o / (v / 0.5e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(cosTheta_O / Float32(v / Float32(0.5))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * (cosTheta_O / (v / single(0.5)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

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 55.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \frac{1}{2}\right), v\right) \]

7. Simplified 55.4%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   2. *-commutative N/A

      \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{\frac{1}{2}}}{v} \]

   3. associate-*l* N/A

      \[\leadsto cosTheta\_i \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{\frac{1}{2}}{v}\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(cosTheta\_O \cdot \frac{\frac{1}{2}}{v}\right)}\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{\frac{1}{2}}{v}\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{\color{blue}{2 \cdot v}}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{v \cdot \color{blue}{2}}\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_O}{\color{blue}{v \cdot 2}}\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, \color{blue}{\left(v \cdot 2\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, \left(v \cdot \frac{1}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   11. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, \left(\frac{v}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   12. /-lowering-/.f32 55.4%

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

9. Applied egg-rr 55.4%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{\frac{v}{0.5}}} \]
10. Add Preprocessing

Alternative 19: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_O (/ cosTheta_i v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_O * (cosTheta_i / 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 = 0.5e0 * (costheta_o * (costheta_i / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_O * Float32(cosTheta_i / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_O * (cosTheta_i / v));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

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 55.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \frac{1}{2}\right), v\right) \]

7. Simplified 55.4%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v} \]

   2. associate-/l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{\mathsf{neg}\left(v\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v\right)} \]

   5. un-div-inv N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(v\right)}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{v}\right)}\right) \]

   7. frac-2neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{-1}{\color{blue}{v}}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot \frac{-1}{v}\right)}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right) \cdot \frac{-1}{v}\right)\right) \]

   10. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{-1}{v}\right)\right)\right) \]

   11. frac-2neg N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   13. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot cosTheta\_i}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   14. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{neg}\left(v\right)}\right)\right)\right) \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\mathsf{neg}\left(cosTheta\_O \cdot \left(\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)\right)\right)\right)\right) \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{cosTheta\_i}{v}\right)\right)\right)\right)}\right)\right) \]

   17. remove-double-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{\color{blue}{v}}\right)\right) \]

   18. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right)\right) \]

   19. /-lowering-/.f32 55.3%

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right)\right) \]

9. Applied egg-rr 55.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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)))