HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.9%

Time: 19.9s

Alternatives: 20

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_O (exp (/ sinTheta_i (/ v sinTheta_O))))
  (* cosTheta_i (/ (/ 1.0 v) (/ (sinh (/ 1.0 v)) (/ 0.5 v))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 / exp((sintheta_i / (v / sintheta_o)))) * (costheta_i * ((1.0e0 / v) / (sinh((1.0e0 / v)) / (0.5e0 / v))))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / exp(Float32(sinTheta_i / Float32(v / sinTheta_O)))) * Float32(cosTheta_i * Float32(Float32(Float32(1.0) / v) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(Float32(0.5) / v)))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / exp((sinTheta_i / (v / sinTheta_O)))) * (cosTheta_i * ((single(1.0) / v) / (sinh((single(1.0) / v)) / (single(0.5) / v))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

   1. div-inv N/A

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

   2. associate-*l* N/A

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

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

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

   11. /-lowering-/.f32 99.1%

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

8. Applied egg-rr 99.1%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_O (exp (/ sinTheta_i (/ v sinTheta_O))))
  (* (/ cosTheta_i v) (/ (/ 0.5 v) (sinh (/ 1.0 v))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 / exp((sintheta_i / (v / sintheta_o)))) * ((costheta_i / v) * ((0.5e0 / v) / sinh((1.0e0 / v))))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / exp(Float32(sinTheta_i / Float32(v / sinTheta_O)))) * Float32(Float32(cosTheta_i / v) * Float32(Float32(Float32(0.5) / v) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / exp((sinTheta_i / (v / sinTheta_O)))) * ((cosTheta_i / v) * ((single(0.5) / v) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (/
   cosTheta_i
   (*
    (exp (/ sinTheta_i (/ v sinTheta_O)))
    (* v (* (* v -2.0) (sinh (/ -1.0 v))))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 / (exp((sintheta_i / (v / sintheta_o))) * (v * ((v * (-2.0e0)) * sinh(((-1.0e0) / v))))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (cosTheta_i / (exp((sinTheta_i / (v / sinTheta_O))) * (v * ((v * single(-2.0)) * sinh((single(-1.0) / v))))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

   1. associate-/l/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 98.8% accurate, 1.6× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* (/ cosTheta_i v) (/ (/ 0.5 v) (sinh (/ 1.0 v))))
  (-
   cosTheta_O
   (/
    (+
     (*
      (/
       (* (* cosTheta_O (* sinTheta_O sinTheta_O)) (* sinTheta_i sinTheta_i))
       v)
      -0.5)
     (* cosTheta_O (* sinTheta_i sinTheta_O)))
    v))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i / v) * ((0.5e0 / v) / sinh((1.0e0 / v)))) * (costheta_o - ((((((costheta_o * (sintheta_o * sintheta_o)) * (sintheta_i * sintheta_i)) / v) * (-0.5e0)) + (costheta_o * (sintheta_i * sintheta_o))) / v))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i / v) * ((single(0.5) / v) / sinh((single(1.0) / v)))) * (cosTheta_O - ((((((cosTheta_O * (sinTheta_O * sinTheta_O)) * (sinTheta_i * sinTheta_i)) / v) * single(-0.5)) + (cosTheta_O * (sinTheta_i * sinTheta_O))) / v));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (* cosTheta_i (/ (/ 1.0 v) (/ (sinh (/ 1.0 v)) (/ 0.5 v))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 * ((1.0e0 / v) / (sinh((1.0e0 / v)) / (0.5e0 / v))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (cosTheta_i * ((single(1.0) / v) / (sinh((single(1.0) / v)) / (single(0.5) / v))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

   1. div-inv N/A

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

   2. associate-*l* N/A

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

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

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

   11. /-lowering-/.f32 99.1%

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

8. Applied egg-rr 99.1%

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

9. Taylor expanded in sinTheta_i around 0

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

11. Simplified 98.9%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (/ (/ cosTheta_i v) (/ (sinh (/ 1.0 v)) (/ 0.5 v)))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o * ((costheta_i / v) / (sinh((1.0e0 / v)) / (0.5e0 / v)))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) / (sinh((single(1.0) / v)) / (single(0.5) / v)));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

    1. clear-num N/A

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

    2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

    8. /-lowering-/.f32 98.8%

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

11. Applied egg-rr 98.8%

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

Alternative 7: 98.5% accurate, 1.9× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (* (/ cosTheta_i v) (/ (/ 0.5 v) (sinh (/ 1.0 v))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o * ((costheta_i / v) * ((0.5e0 / v) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) * ((single(0.5) / v) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

Alternative 8: 98.5% accurate, 1.9× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (* cosTheta_i (/ (/ 0.5 v) (* v (sinh (/ 1.0 v)))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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) / (v * sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (cosTheta_i * ((single(0.5) / v) / (v * sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

    1. frac-times N/A

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

    2. associate-/l* N/A

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

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

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

    4. *-commutative N/A

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

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

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

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

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

    7. *-commutative N/A

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

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

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

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

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

    10. /-lowering-/.f32 98.8%

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

11. Applied egg-rr 98.8%

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

Alternative 9: 70.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := -1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\\ \frac{-0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v \cdot t\_0} + sinTheta\_i \cdot \left(0.5 \cdot \left(sinTheta\_i \cdot \left(-0.5 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{t\_0 \cdot \left(v \cdot \left(v \cdot v\right)\right)}\right) + \frac{\frac{cosTheta\_O \cdot \left(sinTheta\_O \cdot cosTheta\_i\right)}{v \cdot v}}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (-
          -1.0
          (/
           (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v)))
           (* v v)))))
   (+
    (/ (* -0.5 (* cosTheta_O cosTheta_i)) (* v t_0))
    (*
     sinTheta_i
     (*
      0.5
      (+
       (*
        sinTheta_i
        (*
         -0.5
         (/
          (* cosTheta_O (* cosTheta_i (* sinTheta_O sinTheta_O)))
          (* t_0 (* v (* v v))))))
       (/ (/ (* cosTheta_O (* sinTheta_O cosTheta_i)) (* v v)) t_0)))))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = -1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v));
	return ((-0.5f * (cosTheta_O * cosTheta_i)) / (v * t_0)) + (sinTheta_i * (0.5f * ((sinTheta_i * (-0.5f * ((cosTheta_O * (cosTheta_i * (sinTheta_O * sinTheta_O))) / (t_0 * (v * (v * v)))))) + (((cosTheta_O * (sinTheta_O * cosTheta_i)) / (v * v)) / t_0))));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))
    code = (((-0.5e0) * (costheta_o * costheta_i)) / (v * t_0)) + (sintheta_i * (0.5e0 * ((sintheta_i * ((-0.5e0) * ((costheta_o * (costheta_i * (sintheta_o * sintheta_o))) / (t_0 * (v * (v * v)))))) + (((costheta_o * (sintheta_o * costheta_i)) / (v * v)) / t_0))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v));
	tmp = ((single(-0.5) * (cosTheta_O * cosTheta_i)) / (v * t_0)) + (sinTheta_i * (single(0.5) * ((sinTheta_i * (single(-0.5) * ((cosTheta_O * (cosTheta_i * (sinTheta_O * sinTheta_O))) / (t_0 * (v * (v * v)))))) + (((cosTheta_O * (sinTheta_O * cosTheta_i)) / (v * v)) / t_0))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 71.4%

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

8. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)} + sinTheta\_i \cdot \left(\frac{1}{2} \cdot \left(sinTheta\_i \cdot \left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot {sinTheta\_O}^{2}\right)}{{v}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)} + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot {sinTheta\_O}^{2}\right)}{{v}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}\right)\right) + \frac{1}{2} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot sinTheta\_O\right)}{{v}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}\right)} \]

10. Simplified 71.4%

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

11. Final simplification 71.4%

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

Alternative 10: 70.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := -1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\\ \frac{-0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v \cdot t\_0} + \frac{0.5 \cdot \left(cosTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot cosTheta\_i\right)\right)\right)}{\left(v \cdot v\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (-
          -1.0
          (/
           (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v)))
           (* v v)))))
   (+
    (/ (* -0.5 (* cosTheta_O cosTheta_i)) (* v t_0))
    (/
     (* 0.5 (* cosTheta_O (* sinTheta_i (* sinTheta_O cosTheta_i))))
     (* (* v v) t_0)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = -1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v));
	return ((-0.5f * (cosTheta_O * cosTheta_i)) / (v * t_0)) + ((0.5f * (cosTheta_O * (sinTheta_i * (sinTheta_O * cosTheta_i)))) / ((v * v) * t_0));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))
    code = (((-0.5e0) * (costheta_o * costheta_i)) / (v * t_0)) + ((0.5e0 * (costheta_o * (sintheta_i * (sintheta_o * costheta_i)))) / ((v * v) * t_0))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v));
	tmp = ((single(-0.5) * (cosTheta_O * cosTheta_i)) / (v * t_0)) + ((single(0.5) * (cosTheta_O * (sinTheta_i * (sinTheta_O * cosTheta_i)))) / ((v * v) * t_0));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 71.4%

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

8. Taylor expanded in sinTheta_i around 0

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

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

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

10. Simplified 71.3%

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

11. Final simplification 71.3%

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

Alternative 11: 70.2% accurate, 8.8× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (*
   (/ cosTheta_i v)
   (/
    (/ 0.5 v)
    (/
     (+
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      1.0)
     v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i / v) * ((0.5f / v) / ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) + 1.0f) / v)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o * ((costheta_i / v) * ((0.5e0 / v) / ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) + 1.0e0) / v)))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) * ((single(0.5) / v) / ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) + single(1.0)) / v)));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

10. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. mul-1-neg N/A

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

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

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

12. Simplified 71.3%

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

13. Final simplification 71.3%

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

Alternative 12: 70.2% accurate, 10.5× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* -0.5 (* cosTheta_O cosTheta_i))
  (*
   v
   (-
    -1.0
    (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (-0.5f * (cosTheta_O * cosTheta_i)) / (v * (-1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v))));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((-0.5e0) * (costheta_o * costheta_i)) / (v * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(-0.5) * (cosTheta_O * cosTheta_i)) / (v * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 71.4%

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

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 71.2%

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

11. Final simplification 71.2%

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

Alternative 13: 64.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot \frac{\frac{0.5}{v}}{\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v}}\right) \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O
  (*
   (/ cosTheta_i v)
   (/ (/ 0.5 v) (/ (+ (/ 0.16666666666666666 (* v v)) 1.0) v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * ((cosTheta_i / v) * ((0.5f / v) / (((0.16666666666666666f / (v * v)) + 1.0f) / v)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o * ((costheta_i / v) * ((0.5e0 / v) / (((0.16666666666666666e0 / (v * v)) + 1.0e0) / v)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(Float32(cosTheta_i / v) * Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) + Float32(1.0)) / v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * ((cosTheta_i / v) * ((single(0.5) / v) / (((single(0.16666666666666666) / (v * v)) + single(1.0)) / v)));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   16. associate-/l/ N/A

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

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

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Simplified 98.8%

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

10. Taylor expanded in v around inf

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

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

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

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

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

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

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

    6. unpow2 N/A

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

    7. *-lowering-*.f32 64.9%

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

12. Simplified 64.9%

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

13. Final simplification 64.9%

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

Alternative 14: 59.0% accurate, 20.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{-1}{\frac{\frac{2}{cosTheta\_O \cdot cosTheta\_i}}{\frac{-1}{v}}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ -1.0 (/ (/ 2.0 (* cosTheta_O cosTheta_i)) (/ -1.0 v))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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) / ((2.0e0 / (costheta_o * costheta_i)) / ((-1.0e0) / v))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(-1.0) / Float32(Float32(Float32(2.0) / Float32(cosTheta_O * cosTheta_i)) / Float32(Float32(-1.0) / v)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(-1.0) / ((single(2.0) / (cosTheta_O * cosTheta_i)) / (single(-1.0) / v));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

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

   1. clear-num N/A

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

   2. associate-*r* N/A

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

   3. associate-/l* N/A

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

   4. associate-/r* N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   12. associate-/l* N/A

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

   13. associate-/l* N/A

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

   14. associate-/r* N/A

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

6. Applied egg-rr 94.2%

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

7. Taylor expanded in v around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

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

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

   12. *-lowering-*.f32 69.1%

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

9. Simplified 69.1%

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

    1. associate-/l/ N/A

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

    2. clear-num N/A

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

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

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

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

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

11. Applied egg-rr 61.7%

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

12. Taylor expanded in v around inf

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

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

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

    2. *-lowering-*.f32 59.8%

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

14. Simplified 59.8%

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

15. Final simplification 59.8%

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

Alternative 15: 58.8% accurate, 24.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{1}{\frac{\frac{v}{cosTheta\_O \cdot cosTheta\_i}}{0.5}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ (/ v (* cosTheta_O cosTheta_i)) 0.5)))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o * costheta_i)) / 0.5e0)
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(Float32(v / Float32(cosTheta_O * cosTheta_i)) / Float32(0.5)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / ((v / (cosTheta_O * cosTheta_i)) / single(0.5));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 59.1%

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

7. Simplified 59.1%

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

   1. clear-num N/A

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

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

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 59.7%

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

9. Applied egg-rr 59.7%

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

Alternative 16: 58.4% accurate, 24.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{1}{v} \cdot \frac{cosTheta\_O}{\frac{2}{cosTheta\_i}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 1.0 v) (/ cosTheta_O (/ 2.0 cosTheta_i))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (1.0f / v) * (cosTheta_O / (2.0f / cosTheta_i));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o / (2.0e0 / costheta_i))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(1.0) / v) * (cosTheta_O / (single(2.0) / cosTheta_i));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 59.1%

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

7. Simplified 59.1%

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

   1. div-inv N/A

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

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. /-lowering-/.f32 59.1%

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

9. Applied egg-rr 59.1%

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

10. Final simplification 59.1%

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

Alternative 17: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{\frac{cosTheta\_O}{\frac{2}{cosTheta\_i}}}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ cosTheta_O (/ 2.0 cosTheta_i)) v))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / (2.0f / cosTheta_i)) / v;
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 / (2.0e0 / costheta_i)) / v
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / Float32(Float32(2.0) / cosTheta_i)) / v)
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / (single(2.0) / cosTheta_i)) / v;
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 59.1%

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

7. Simplified 59.1%

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

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

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

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

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

   10. /-lowering-/.f32 59.1%

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

9. Applied egg-rr 59.1%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{2}{cosTheta\_i}}}{v}} \]
10. Add Preprocessing

Alternative 18: 58.4% accurate, 31.4× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_i (* cosTheta_O 0.5)) v))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(cosTheta_O * Float32(0.5))) / v)
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
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 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 59.1%

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

7. Simplified 59.1%

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

Alternative 19: 58.3% accurate, 31.4× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* cosTheta_O (/ cosTheta_i v))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * (cosTheta_O * (cosTheta_i / v));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 * (costheta_o * (costheta_i / v))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(cosTheta_O * Float32(cosTheta_i / v)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (cosTheta_O * (cosTheta_i / v));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 59.1%

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

7. Simplified 59.1%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
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 59.1%

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

9. Applied egg-rr 59.1%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

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

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

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

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

    5. /-lowering-/.f32 59.1%

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

11. Applied egg-rr 59.1%

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

12. Final simplification 59.1%

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

Alternative 20: 58.3% accurate, 31.4× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ cosTheta_i v) (* cosTheta_O 0.5)))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i / v) * (cosTheta_O * 0.5f);
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i / v) * (costheta_o * 0.5e0)
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i / v) * Float32(cosTheta_O * Float32(0.5)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i / v) * (cosTheta_O * single(0.5));
end

Derivation

1. Initial program 98.9%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
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 59.1%

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

7. Simplified 59.1%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
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 59.1%

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

9. Applied egg-rr 59.1%

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

10. Final simplification 59.1%

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

Reproduce

herbie shell --seed 2024149 
(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)))