HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 20.7s

Alternatives: 21

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

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))) (exp (/ sinTheta_i (/ v sinTheta_O)))))))

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))) / expf((sinTheta_i / (v / sinTheta_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 = (costheta_i * costheta_o) * ((1.0e0 / v) * (((0.5e0 / v) / sinh((1.0e0 / v))) / exp((sintheta_i / (v / sintheta_o)))))
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))) / exp(Float32(sinTheta_i / Float32(v / sinTheta_O))))))
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))) / exp((sinTheta_i / (v / sinTheta_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. div-inv N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{1}{\frac{2}{\sinh \left(\frac{1}{v}\right) \cdot 2}}}}{-1 - sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666}{v \cdot v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (*
   (/ -1.0 v)
   (/
    (/ (/ 0.5 v) (/ 1.0 (/ 2.0 (* (sinh (/ 1.0 v)) 2.0))))
    (-
     -1.0
     (*
      sinTheta_i
      (+
       (/ sinTheta_O v)
       (*
        sinTheta_i
        (+
         (*
          (/ 0.16666666666666666 (* v v))
          (/ (* sinTheta_i (* sinTheta_O (* sinTheta_O sinTheta_O))) v))
         (/ (* 0.5 (* sinTheta_O sinTheta_O)) (* 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) / (1.0f / (2.0f / (sinhf((1.0f / v)) * 2.0f)))) / (-1.0f - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((0.16666666666666666f / (v * v)) * ((sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O))) / v)) + ((0.5f * (sinTheta_O * sinTheta_O)) / (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) / (1.0e0 / (2.0e0 / (sinh((1.0e0 / v)) * 2.0e0)))) / ((-1.0e0) - (sintheta_i * ((sintheta_o / v) + (sintheta_i * (((0.16666666666666666e0 / (v * v)) * ((sintheta_i * (sintheta_o * (sintheta_o * sintheta_o))) / v)) + ((0.5e0 * (sintheta_o * sintheta_o)) / (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) / Float32(Float32(1.0) / Float32(Float32(2.0) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))) / Float32(Float32(-1.0) - Float32(sinTheta_i * Float32(Float32(sinTheta_O / v) + Float32(sinTheta_i * Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) * Float32(Float32(sinTheta_i * Float32(sinTheta_O * Float32(sinTheta_O * sinTheta_O))) / v)) + Float32(Float32(Float32(0.5) * Float32(sinTheta_O * sinTheta_O)) / 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) / (single(1.0) / (single(2.0) / (sinh((single(1.0) / v)) * single(2.0))))) / (single(-1.0) - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((single(0.16666666666666666) / (v * v)) * ((sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O))) / v)) + ((single(0.5) * (sinTheta_O * sinTheta_O)) / (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

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

   1. sinh-def N/A

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

   2. sinh-undef N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f32 98.9%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in sinTheta_i around 0

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

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

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

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

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

    3. +-commutative N/A

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

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

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

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

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

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

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

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

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

11. Simplified 98.9%

    \[\leadsto \left(-cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{1}{\frac{2}{\sinh \left(\frac{1}{v}\right) \cdot 2}}}}{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666}{v \cdot v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}\right) \]

12. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

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 / v) / sinhf((1.0f / v)))) / (-1.0f - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((0.5f * (sinTheta_O * sinTheta_O)) / (v * v)) + ((0.16666666666666666f / (v * v)) / (v / (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_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 = (((costheta_i * costheta_o) / v) * (((-0.5e0) / v) / sinh((1.0e0 / v)))) / ((-1.0e0) - (sintheta_i * ((sintheta_o / v) + (sintheta_i * (((0.5e0 * (sintheta_o * sintheta_o)) / (v * v)) + ((0.16666666666666666e0 / (v * v)) / (v / (sintheta_i * (sintheta_o * (sintheta_o * sintheta_o))))))))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * ((single(-0.5) / v) / sinh((single(1.0) / v)))) / (single(-1.0) - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((single(0.5) * (sinTheta_O * sinTheta_O)) / (v * v)) + ((single(0.16666666666666666) / (v * v)) / (v / (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_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. div-inv N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

   1. sinh-def N/A

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

   2. sinh-undef N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f32 98.9%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in sinTheta_i around 0

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

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

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

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

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

    3. +-commutative N/A

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

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

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

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

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

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

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

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

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

11. Simplified 98.9%

    \[\leadsto \left(-cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{1}{\frac{2}{\sinh \left(\frac{1}{v}\right) \cdot 2}}}}{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666}{v \cdot v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}\right) \]

13. Applied egg-rr 98.9%

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

14. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ 0.5 v) (/ (* cosTheta_i cosTheta_O) v))
  (*
   (sinh (/ 1.0 v))
   (+
    1.0
    (*
     sinTheta_i
     (+
      (/ sinTheta_O v)
      (*
       sinTheta_i
       (+
        (/ (* 0.5 (* sinTheta_O sinTheta_O)) (* v v))
        (/
         (/ 0.16666666666666666 (* v v))
         (/ v (* sinTheta_i (* sinTheta_O (* sinTheta_O sinTheta_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) / v)) / (sinhf((1.0f / v)) * (1.0f + (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((0.5f * (sinTheta_O * sinTheta_O)) / (v * v)) + ((0.16666666666666666f / (v * v)) / (v / (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_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) / v)) / (sinh((1.0e0 / v)) * (1.0e0 + (sintheta_i * ((sintheta_o / v) + (sintheta_i * (((0.5e0 * (sintheta_o * sintheta_o)) / (v * v)) + ((0.16666666666666666e0 / (v * v)) / (v / (sintheta_i * (sintheta_o * (sintheta_o * sintheta_o)))))))))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((single(0.5) / v) * ((cosTheta_i * cosTheta_O) / v)) / (sinh((single(1.0) / v)) * (single(1.0) + (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * (((single(0.5) * (sinTheta_O * sinTheta_O)) / (v * v)) + ((single(0.16666666666666666) / (v * v)) / (v / (sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_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. div-inv N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

   1. sinh-def N/A

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

   2. sinh-undef N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f32 98.9%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in sinTheta_i around 0

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

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

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

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

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

    3. +-commutative N/A

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

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

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

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

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

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

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

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

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

11. Simplified 98.9%

    \[\leadsto \left(-cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{1}{\frac{2}{\sinh \left(\frac{1}{v}\right) \cdot 2}}}}{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666}{v \cdot v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}\right) \]

13. Applied egg-rr 98.8%

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

14. Final simplification 98.8%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_i cosTheta_O) (/ (/ -1.0 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_i * cosTheta_O) * ((-1.0f / 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_i * costheta_o) * (((-1.0e0) / v) / (sinh((1.0e0 / v)) / ((-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(Float32(-1.0) / 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_i * cosTheta_O) * ((single(-1.0) / 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. div-inv N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

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

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

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

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

   9. rec-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

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

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

   13. /-lowering-/.f32 98.1%

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

9. Simplified 98.1%

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

    1. associate-/r* N/A

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

    2. associate-/l/ N/A

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

    3. div-inv N/A

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

    4. neg-mul-1 N/A

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

    5. sinh-undef N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. frac-2neg N/A

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

    10. distribute-neg-frac N/A

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

    11. metadata-eval N/A

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

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

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

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

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

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

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

Alternative 6: 98.6% accurate, 1.9× speedup

Math

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

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))))))

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))));
}

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))))
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(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(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. clear-num N/A

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

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

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

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

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

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

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

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

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

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

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

   10. associate-/l* N/A

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

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

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

   12. /-lowering-/.f32 95.5%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in sinTheta_i around 0

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

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

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

   2. *-lowering-*.f32 95.1%

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

9. Simplified 95.1%

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

    1. div-inv N/A

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

    2. *-commutative N/A

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

    3. clear-num N/A

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

    4. div-inv N/A

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

    5. associate-*l* N/A

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

    6. sinh-def N/A

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

    7. sinh-undef N/A

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

    8. *-commutative N/A

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

    9. clear-num N/A

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

    10. metadata-eval N/A

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

    11. div-inv N/A

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

    12. clear-num N/A

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

    13. div-inv N/A

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

    14. clear-num N/A

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

11. Applied egg-rr 98.7%

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

Alternative 7: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

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

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

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

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

   9. rec-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

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

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

   13. /-lowering-/.f32 98.1%

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

9. Simplified 98.1%

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

    1. *-commutative N/A

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

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

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

    3. associate-*r* N/A

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

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

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

11. Applied egg-rr 98.5%

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

12. Final simplification 98.5%

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

Alternative 8: 98.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ (* cosTheta_i cosTheta_O) 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_i * cosTheta_O) / 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_i * costheta_o) / v) / (sinh((1.0e0 / v)) / (0.5e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / 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_i * cosTheta_O) / 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. clear-num N/A

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

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

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

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

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

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

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

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

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

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

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

   10. associate-/l* N/A

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

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

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

   12. /-lowering-/.f32 95.5%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in sinTheta_i around 0

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

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

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

   2. *-lowering-*.f32 95.1%

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

9. Simplified 95.1%

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

    1. *-commutative N/A

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

    2. clear-num N/A

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

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

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

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

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

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

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

    6. sinh-def N/A

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

    7. sinh-undef N/A

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

    8. *-commutative N/A

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

    9. clear-num N/A

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

    10. metadata-eval N/A

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

    11. div-inv N/A

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

    12. clear-num N/A

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

    13. div-inv N/A

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

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

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

    15. clear-num N/A

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

    16. *-commutative N/A

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

    17. sinh-undef N/A

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

    18. sinh-def N/A

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

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

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

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

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

    21. /-lowering-/.f32 98.5%

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

11. Applied egg-rr 98.5%

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

Alternative 9: 98.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* cosTheta_i cosTheta_O) (/ (/ 0.5 v) (* 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) * ((0.5f / v) / (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) * ((0.5e0 / v) / (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(0.5) / v) / Float32(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(0.5) / v) / (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

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-/r* N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

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

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

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

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

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

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

   9. rec-exp N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

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

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

   13. /-lowering-/.f32 98.1%

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

9. Simplified 98.1%

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

    1. *-commutative N/A

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

    2. distribute-rgt-neg-out N/A

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

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

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

    4. div-inv N/A

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

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

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

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

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

11. Applied egg-rr 98.5%

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

12. Final simplification 98.5%

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

Alternative 10: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * (((single(0.5) / v) / sinh((single(1.0) / v))) * (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. clear-num N/A

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

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

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

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

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

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

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

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

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

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

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

   10. associate-/l* N/A

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

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

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

   12. /-lowering-/.f32 95.5%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in sinTheta_i around 0

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

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

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

   2. *-lowering-*.f32 95.1%

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

9. Simplified 95.1%

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

    1. div-inv N/A

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

    2. *-commutative N/A

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

    3. clear-num N/A

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

    4. associate-/l* N/A

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

    5. associate-*l* N/A

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

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

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

    7. sinh-def N/A

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

    8. sinh-undef N/A

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

    9. *-commutative N/A

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

    10. clear-num N/A

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

    11. metadata-eval N/A

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

    12. div-inv N/A

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

    13. clear-num N/A

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

    14. div-inv N/A

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

    15. clear-num N/A

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

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

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

11. Applied egg-rr 98.5%

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

12. Final simplification 98.5%

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

Alternative 11: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.16666666666666666}{v \cdot v}\\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{-1}{\frac{2}{2 \cdot \frac{\left(-1 - t\_0\right) - \frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}}}}}{-1 - sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(t\_0 \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ 0.16666666666666666 (* v v))))
   (*
    (* cosTheta_i cosTheta_O)
    (*
     (/ -1.0 v)
     (/
      (/
       (/ 0.5 v)
       (/
        -1.0
        (/
         2.0
         (*
          2.0
          (/
           (-
            (- -1.0 t_0)
            (/
             (+ 0.008333333333333333 (/ 0.0001984126984126984 (* v v)))
             (* (* v v) (* v v))))
           v)))))
      (-
       -1.0
       (*
        sinTheta_i
        (+
         (/ sinTheta_O v)
         (*
          sinTheta_i
          (+
           (*
            t_0
            (/ (* sinTheta_i (* sinTheta_O (* sinTheta_O sinTheta_O))) v))
           (/ (* 0.5 (* sinTheta_O sinTheta_O)) (* v v))))))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 0.16666666666666666f / (v * v);
	return (cosTheta_i * cosTheta_O) * ((-1.0f / v) * (((0.5f / v) / (-1.0f / (2.0f / (2.0f * (((-1.0f - t_0) - ((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / ((v * v) * (v * v)))) / v))))) / (-1.0f - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * ((t_0 * ((sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O))) / v)) + ((0.5f * (sinTheta_O * sinTheta_O)) / (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
    real(4) :: t_0
    t_0 = 0.16666666666666666e0 / (v * v)
    code = (costheta_i * costheta_o) * (((-1.0e0) / v) * (((0.5e0 / v) / ((-1.0e0) / (2.0e0 / (2.0e0 * ((((-1.0e0) - t_0) - ((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / ((v * v) * (v * v)))) / v))))) / ((-1.0e0) - (sintheta_i * ((sintheta_o / v) + (sintheta_i * ((t_0 * ((sintheta_i * (sintheta_o * (sintheta_o * sintheta_o))) / v)) + ((0.5e0 * (sintheta_o * sintheta_o)) / (v * v)))))))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.16666666666666666) / Float32(v * v))
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(Float32(-1.0) / v) * Float32(Float32(Float32(Float32(0.5) / v) / Float32(Float32(-1.0) / Float32(Float32(2.0) / Float32(Float32(2.0) * Float32(Float32(Float32(Float32(-1.0) - t_0) - Float32(Float32(Float32(0.008333333333333333) + Float32(Float32(0.0001984126984126984) / Float32(v * v))) / Float32(Float32(v * v) * Float32(v * v)))) / v))))) / Float32(Float32(-1.0) - Float32(sinTheta_i * Float32(Float32(sinTheta_O / v) + Float32(sinTheta_i * Float32(Float32(t_0 * Float32(Float32(sinTheta_i * Float32(sinTheta_O * Float32(sinTheta_O * sinTheta_O))) / v)) + Float32(Float32(Float32(0.5) * Float32(sinTheta_O * sinTheta_O)) / Float32(v * v))))))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(0.16666666666666666) / (v * v);
	tmp = (cosTheta_i * cosTheta_O) * ((single(-1.0) / v) * (((single(0.5) / v) / (single(-1.0) / (single(2.0) / (single(2.0) * (((single(-1.0) - t_0) - ((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / ((v * v) * (v * v)))) / v))))) / (single(-1.0) - (sinTheta_i * ((sinTheta_O / v) + (sinTheta_i * ((t_0 * ((sinTheta_i * (sinTheta_O * (sinTheta_O * sinTheta_O))) / v)) + ((single(0.5) * (sinTheta_O * sinTheta_O)) / (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

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. remove-double-neg N/A

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

   5. div-inv N/A

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

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

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

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

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.0%

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

   1. sinh-def N/A

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

   2. sinh-undef N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f32 98.9%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in sinTheta_i around 0

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

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

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

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

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

    3. +-commutative N/A

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

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

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

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

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

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

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

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

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

11. Simplified 98.9%

    \[\leadsto \left(-cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{1}{\frac{2}{\sinh \left(\frac{1}{v}\right) \cdot 2}}}}{\color{blue}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666}{v \cdot v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}}\right) \]

12. Taylor expanded in v around -inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}, 2\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right)\right), v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right)\right), v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{\mathsf{neg}\left(v\right)}\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right)\right), v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{-1 \cdot v}\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right)\right), v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(-1, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right), 2\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{/.f32}\left(sinTheta\_O, v\right), \mathsf{*.f32}\left(sinTheta\_i, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right)\right), v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_O\right)\right), \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

14. Simplified 78.0%

    \[\leadsto \left(-cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{1}{\frac{2}{\color{blue}{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{-\left(v \cdot v\right) \cdot \left(v \cdot v\right)} - \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{-v}} \cdot 2}}}}{1 + sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666}{v \cdot v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}\right) \]

15. Final simplification 78.0%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{-1}{v} \cdot \frac{\frac{\frac{0.5}{v}}{\frac{-1}{\frac{2}{2 \cdot \frac{\left(-1 - \frac{0.16666666666666666}{v \cdot v}\right) - \frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v}}}}}{-1 - sinTheta\_i \cdot \left(\frac{sinTheta\_O}{v} + sinTheta\_i \cdot \left(\frac{0.16666666666666666}{v \cdot v} \cdot \frac{sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v} + \frac{0.5 \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)}{v \cdot v}\right)\right)}\right) \]
16. Add Preprocessing

Alternative 12: 74.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}{\frac{\left(1 + \frac{0.16666666666666666}{v \cdot v}\right) + \frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v} \cdot \left(v \cdot 2\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ 1.0 (/ v (* cosTheta_i cosTheta_O)))
  (*
   (/
    (+
     (+ 1.0 (/ 0.16666666666666666 (* v v)))
     (/
      (+ 0.008333333333333333 (/ 0.0001984126984126984 (* v v)))
      (* (* v v) (* v v))))
    v)
   (* v 2.0))))

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))) / ((((1.0f + (0.16666666666666666f / (v * v))) + ((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / ((v * v) * (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 = (1.0e0 / (v / (costheta_i * costheta_o))) / ((((1.0e0 + (0.16666666666666666e0 / (v * v))) + ((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / ((v * v) * (v * v)))) / v) * (v * 2.0e0))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(1.0) / Float32(v / Float32(cosTheta_i * cosTheta_O))) / Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v))) + Float32(Float32(Float32(0.008333333333333333) + Float32(Float32(0.0001984126984126984) / Float32(v * v))) / Float32(Float32(v * v) * Float32(v * v)))) / v) * Float32(v * Float32(2.0))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) / (v / (cosTheta_i * cosTheta_O))) / ((((single(1.0) + (single(0.16666666666666666) / (v * v))) + ((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / ((v * v) * (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. Step-by-step derivation

   1. clear-num N/A

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

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

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

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

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

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

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

   5. associate-/l* N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

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

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

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

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

   10. associate-/l* N/A

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

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

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

   12. /-lowering-/.f32 95.5%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in sinTheta_i around 0

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

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

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

   2. *-lowering-*.f32 95.1%

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

9. Simplified 95.1%

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

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right)\right), \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{\mathsf{neg}\left(v\right)}\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

    3. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{-1 \cdot v}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right), \mathsf{*.f32}\left(\color{blue}{v}, 2\right)\right)\right) \]

12. Simplified 76.0%

    \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}}}{\color{blue}{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{-\left(v \cdot v\right) \cdot \left(v \cdot v\right)} - \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{-v}} \cdot \left(v \cdot 2\right)} \]

13. Final simplification 76.0%

    \[\leadsto \frac{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}{\frac{\left(1 + \frac{0.16666666666666666}{v \cdot v}\right) + \frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}}{v} \cdot \left(v \cdot 2\right)} \]
14. Add Preprocessing

Alternative 13: 70.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v}\right)}{v}}{\left(v \cdot 2\right) \cdot \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/
   (* cosTheta_O (- cosTheta_i (/ (* cosTheta_i (* sinTheta_i sinTheta_O)) v)))
   v)
  (*
   (* v 2.0)
   (/
    (-
     (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
     -1.0)
    v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * (cosTheta_i - ((cosTheta_i * (sinTheta_i * sinTheta_O)) / v))) / v) / ((v * 2.0f) * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * 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_o * (costheta_i - ((costheta_i * (sintheta_i * sintheta_o)) / v))) / v) / ((v * 2.0e0) * ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_i - Float32(Float32(cosTheta_i * Float32(sinTheta_i * sinTheta_O)) / v))) / v) / Float32(Float32(v * Float32(2.0)) * Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) - Float32(-1.0)) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O * (cosTheta_i - ((cosTheta_i * (sinTheta_i * sinTheta_O)) / v))) / v) / ((v * single(2.0)) * ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - single(-1.0)) / v));
end

Derivation

1. Initial program 98.6%

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 71.1%

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

8. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 71.1%

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

10. Simplified 71.1%

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

11. Final simplification 71.1%

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

Alternative 14: 70.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{-0.5}{v \cdot \left(-1 - \frac{0.16666666666666666 + \frac{\frac{0.008333333333333333}{v}}{v}}{v \cdot v}\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i
  (*
   cosTheta_O
   (/
    -0.5
    (*
     v
     (-
      -1.0
      (/
       (+ 0.16666666666666666 (/ (/ 0.008333333333333333 v) v))
       (* v 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 * (-1.0f - ((0.16666666666666666f + ((0.008333333333333333f / v) / 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 * (costheta_o * ((-0.5e0) / (v * ((-1.0e0) - ((0.16666666666666666e0 + ((0.008333333333333333e0 / v) / v)) / (v * v))))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(cosTheta_O * Float32(Float32(-0.5) / Float32(v * Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(Float32(0.008333333333333333) / v) / v)) / Float32(v * v)))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * (cosTheta_O * (single(-0.5) / (v * (single(-1.0) - ((single(0.16666666666666666) + ((single(0.008333333333333333) / v) / 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. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 71.1%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

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

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

10. Simplified 71.0%

    \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot -0.5}{v \cdot \left(\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{0 - v \cdot v} + -1\right)}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}} \]

    2. *-commutative N/A

       \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{\frac{-1}{2}}}{v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)} \]

    3. associate-*l* N/A

       \[\leadsto cosTheta\_i \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{\frac{-1}{2}}{v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}\right)} \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(cosTheta\_O \cdot \frac{\frac{-1}{2}}{v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{\frac{-1}{2}}{v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{-1}{2}, \color{blue}{\left(v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}\right)\right)\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(v, \left(-1 + \color{blue}{\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(-1, \color{blue}{\left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v}\right)}\right)\right)\right)\right)\right) \]

    10. sub0-neg N/A

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

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

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

12. Applied egg-rr 71.1%

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

13. Final simplification 71.1%

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

Alternative 15: 70.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ cosTheta\_O \cdot \frac{cosTheta\_i \cdot -0.5}{v \cdot \left(-1 - \frac{0.16666666666666666 + \frac{\frac{0.008333333333333333}{v}}{v}}{v \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (/
   (* cosTheta_i -0.5)
   (*
    v
    (-
     -1.0
     (/ (+ 0.16666666666666666 (/ (/ 0.008333333333333333 v) v)) (* v v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i * -0.5f) / (v * (-1.0f - ((0.16666666666666666f + ((0.008333333333333333f / v) / 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_o * ((costheta_i * (-0.5e0)) / (v * ((-1.0e0) - ((0.16666666666666666e0 + ((0.008333333333333333e0 / v) / v)) / (v * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(cosTheta_i * Float32(-0.5)) / Float32(v * Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(Float32(0.008333333333333333) / v) / v)) / Float32(v * v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i * single(-0.5)) / (v * (single(-1.0) - ((single(0.16666666666666666) + ((single(0.008333333333333333) / v) / 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. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 71.1%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

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

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

10. Simplified 71.0%

    \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot -0.5}{v \cdot \left(\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{0 - v \cdot v} + -1\right)}} \]
11. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{-1}{2}\right)}{\color{blue}{v} \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)} \]

    2. associate-/l* N/A

       \[\leadsto cosTheta\_O \cdot \color{blue}{\frac{cosTheta\_i \cdot \frac{-1}{2}}{v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}} \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i \cdot \frac{-1}{2}}{v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \frac{-1}{2}\right), \color{blue}{\left(v \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{-1}{2}\right), \left(\color{blue}{v} \cdot \left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{-1}{2}\right), \mathsf{*.f32}\left(v, \color{blue}{\left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v} + -1\right)}\right)\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{-1}{2}\right), \mathsf{*.f32}\left(v, \left(-1 + \color{blue}{\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \frac{-1}{2}\right), \mathsf{*.f32}\left(v, \mathsf{+.f32}\left(-1, \color{blue}{\left(\frac{\frac{1}{6} + \frac{\frac{1}{120}}{v \cdot v}}{0 - v \cdot v}\right)}\right)\right)\right)\right) \]

    9. sub0-neg N/A

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

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

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

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

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

    12. associate-/r* N/A

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

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

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

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

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

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

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

    16. *-lowering-*.f32 71.1%

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

12. Applied egg-rr 71.1%

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

13. Final simplification 71.1%

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

Alternative 16: 63.9% accurate, 14.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* cosTheta_i cosTheta_O) -0.5)
  (* v (+ -1.0 (/ -0.16666666666666666 (* v 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 * (-1.0f + (-0.16666666666666666f / (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) * (-0.5e0)) / (v * ((-1.0e0) + ((-0.16666666666666666e0) / (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(-0.5)) / Float32(v * Float32(Float32(-1.0) + Float32(Float32(-0.16666666666666666) / Float32(v * v)))))
end

MATLAB

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

7. Simplified 71.1%

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

8. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

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

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

10. Simplified 71.0%

    \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot -0.5}{v \cdot \left(\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{0 - v \cdot v} + -1\right)}} \]

11. Taylor expanded in v around inf

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

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

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

    2. unpow2 N/A

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

    3. *-lowering-*.f32 64.9%

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

13. Simplified 64.9%

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

14. Final simplification 64.9%

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

Alternative 17: 58.8% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{v}}{\frac{2}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) / v) / (single(2.0) / (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. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 59.3%

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

7. Simplified 59.3%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. /-lowering-/.f32 59.3%

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

9. Applied egg-rr 59.3%

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

    1. div-inv N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

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

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

    8. *-commutative N/A

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

    9. remove-double-div N/A

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

    10. metadata-eval N/A

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

    11. frac-2neg N/A

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

    12. associate-*l/ N/A

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

    13. div-inv N/A

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

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

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

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

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

    16. *-commutative N/A

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

    17. associate-/r* N/A

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

    18. metadata-eval N/A

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

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

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

    20. *-lowering-*.f32 60.2%

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

11. Applied egg-rr 60.2%

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

Alternative 18: 58.8% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{v \cdot \frac{2}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v * (single(2.0) / (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. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 59.3%

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

7. Simplified 59.3%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. /-lowering-/.f32 59.3%

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

9. Applied egg-rr 59.3%

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

    1. div-inv N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

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

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

    8. *-commutative N/A

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

    9. remove-double-div N/A

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

    10. metadata-eval N/A

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

    11. frac-2neg N/A

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

    12. frac-times N/A

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

    13. metadata-eval N/A

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

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

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

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

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

    16. *-commutative N/A

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

    17. associate-/r* N/A

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

    18. metadata-eval N/A

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

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

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

    20. *-lowering-*.f32 60.2%

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

11. Applied egg-rr 60.2%

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

12. Final simplification 60.2%

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

Alternative 19: 58.6% 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. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 59.3%

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

7. Simplified 59.3%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. /-lowering-/.f32 59.3%

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

9. Applied egg-rr 59.3%

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

    1. *-commutative N/A

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

    2. associate-*l* N/A

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

    3. associate-/l* N/A

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

    4. associate-*l/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\frac{cosTheta\_O}{v} \cdot \color{blue}{cosTheta\_i}\right) \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}\right) \]

    6. /-rgt-identity N/A

       \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{\color{blue}{1}} \]

    7. clear-num N/A

       \[\leadsto \frac{1}{2} \cdot \frac{1}{\color{blue}{\frac{1}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}} \]

    8. un-div-inv N/A

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{1}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}}} \]

    9. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}\right)}\right) \]

    10. associate-*r/ N/A

        \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \left(\frac{1}{\frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v}}}\right)\right) \]

    11. clear-num N/A

        \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \left(\frac{v}{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}\right)\right) \]

    12. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)\right) \]

    13. *-lowering-*.f32 60.0%

        \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{cosTheta\_O}\right)\right)\right) \]

11. Applied egg-rr 60.0%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]
12. Add Preprocessing

Alternative 20: 58.2% 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. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 59.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 59.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

   5. /-lowering-/.f32 59.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 59.3%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{0.5}{v}} \]
10. Add Preprocessing

Alternative 21: 58.2% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta\_O}{\frac{v}{cosTheta\_i}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ cosTheta_O (/ v cosTheta_i))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_O / (v / cosTheta_i));
}

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 / (v / costheta_i))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_O / Float32(v / cosTheta_i)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_O / (v / cosTheta_i));
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. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 59.3%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 59.3%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot cosTheta\_i}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{cosTheta\_i}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot \frac{1}{2}\right), \color{blue}{\left(\frac{cosTheta\_i}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \left(\frac{\color{blue}{cosTheta\_i}}{v}\right)\right) \]

   5. /-lowering-/.f32 59.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 59.3%

   \[\leadsto \color{blue}{\left(cosTheta\_O \cdot 0.5\right) \cdot \frac{cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{cosTheta\_i}}{v} \]

    2. associate-*l* N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)} \]

    3. associate-/l* N/A

       \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{\color{blue}{v}} \]

    4. *-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v} \]

    5. associate-*r/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_O}{v} \cdot \color{blue}{cosTheta\_i}\right)\right) \]

    8. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\left(cosTheta\_O \cdot \frac{1}{v}\right) \cdot cosTheta\_i\right)\right) \]

    9. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \color{blue}{\left(\frac{1}{v} \cdot cosTheta\_i\right)}\right)\right) \]

    10. associate-/r/ N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(cosTheta\_O \cdot \frac{1}{\color{blue}{\frac{v}{cosTheta\_i}}}\right)\right) \]

    11. un-div-inv N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \left(\frac{cosTheta\_O}{\color{blue}{\frac{v}{cosTheta\_i}}}\right)\right) \]

    12. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{v}{cosTheta\_i}\right)}\right)\right) \]

    13. /-lowering-/.f32 59.3%

        \[\leadsto \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \color{blue}{cosTheta\_i}\right)\right)\right) \]

11. Applied egg-rr 59.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_O}{\frac{v}{cosTheta\_i}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))