HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 21.1s

Alternatives: 19

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. frac-2neg N/A

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

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

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

   3. associate-/l* N/A

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

   4. distribute-neg-frac2 N/A

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

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

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

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

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

   7. div-inv N/A

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

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

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

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

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((0.5e0 / v) / sinh((1.0e0 / v))) * (costheta_o / exp((sintheta_i / (v / sintheta_o))))) * (costheta_i / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. associate-*r* N/A

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

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

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

6. Applied egg-rr 98.9%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_o / v) * (costheta_i - (costheta_i * ((sintheta_i * sintheta_o) / v)))) / (sinh((1.0e0 / v)) / (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

7. Simplified 98.5%

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

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

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

   2. associate-/l* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

9. Applied egg-rr 98.7%

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

    1. associate-/r/ N/A

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

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

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

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

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

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

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

    5. div-inv N/A

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

    6. clear-num N/A

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

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

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

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

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

    9. *-commutative N/A

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

    10. *-lowering-*.f32 98.8%

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

11. Applied egg-rr 98.8%

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

12. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o / ((v / costheta_i) + ((sintheta_i * sintheta_o) / costheta_i))) / (sinh((1.0e0 / v)) / (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

7. Simplified 98.5%

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

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

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

   2. associate-/l* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

9. Applied egg-rr 98.7%

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

10. Taylor expanded in sinTheta_i around 0

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

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

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

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

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

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

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

    4. *-lowering-*.f32 98.7%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * ((costheta_o / sinh((1.0e0 / v))) / ((v / costheta_i) + ((sintheta_i * sintheta_o) / costheta_i)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. frac-2neg N/A

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

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

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

   3. associate-/l* N/A

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

   4. distribute-neg-frac2 N/A

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

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

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

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

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

   7. div-inv N/A

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

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

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

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

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. frac-2neg N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. associate-*l/ N/A

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

   11. associate-/r/ N/A

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

   12. *-commutative N/A

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

   13. frac-times N/A

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

   14. associate-/r* N/A

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

8. Applied egg-rr 98.7%

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

9. Taylor expanded in sinTheta_i around 0

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

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

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

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

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

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

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

    4. *-lowering-*.f32 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * (costheta_o / v)) / (sinh((1.0e0 / v)) / (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

7. Simplified 98.5%

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

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

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

   2. associate-/l* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

9. Applied egg-rr 98.7%

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

    1. associate-/r/ N/A

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

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

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

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

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

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

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

    5. div-inv N/A

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

    6. clear-num N/A

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

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

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

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

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

    9. *-commutative N/A

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

    10. *-lowering-*.f32 98.8%

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

11. Applied egg-rr 98.8%

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

12. Taylor expanded in sinTheta_O around 0

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

14. Simplified 98.6%

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

15. Final simplification 98.6%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * ((costheta_o / sinh((1.0e0 / v))) / (v / costheta_i))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. frac-2neg N/A

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

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

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

   3. associate-/l* N/A

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

   4. distribute-neg-frac2 N/A

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

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

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

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

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

   7. div-inv N/A

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

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

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

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

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. frac-2neg N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. associate-*l/ N/A

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

   11. associate-/r/ N/A

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

   12. *-commutative N/A

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

   13. frac-times N/A

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

   14. associate-/r* N/A

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

8. Applied egg-rr 98.7%

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

9. Taylor expanded in sinTheta_i around 0

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

    1. /-lowering-/.f32 98.4%

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

11. Simplified 98.4%

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

12. Final simplification 98.4%

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

Alternative 8: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o / sinh((1.0e0 / v))) * ((costheta_i * 0.5e0) / (v * v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. associate-/l* N/A

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

   16. clear-num N/A

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

   17. un-div-inv N/A

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

   18. /-lowering-/.f32 N/A

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

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

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

   20. *-lowering-*.f32 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f32 98.3%

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

9. Simplified 98.3%

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

Alternative 9: 70.7% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o / ((v / (costheta_i - (costheta_i * ((sintheta_i * sintheta_o) / v)))) / ((0.5e0 / v) / ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) + 1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O / Float32(Float32(v / Float32(cosTheta_i - Float32(cosTheta_i * Float32(Float32(sinTheta_i * sinTheta_O) / 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

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

7. Simplified 98.5%

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

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

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

   2. associate-/l* N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

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

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

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

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

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

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

   15. metadata-eval N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

9. Applied egg-rr 98.7%

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

    1. associate-/l/ N/A

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

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

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

    3. clear-num N/A

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

    4. associate-*l/ N/A

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

    5. clear-num N/A

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

    6. div-inv N/A

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

    7. clear-num N/A

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

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

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

11. Applied egg-rr 98.6%

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

12. Taylor expanded in v around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. mul-1-neg N/A

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

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

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

14. Simplified 69.8%

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

15. Final simplification 69.8%

    \[\leadsto \frac{cosTheta\_O}{\frac{\frac{v}{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\frac{\frac{0.5}{v}}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} + 1}{v}}}} \]
16. Add Preprocessing

Alternative 10: 70.7% accurate, 6.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O * (cosTheta_i - ((cosTheta_i * (sinTheta_i * sinTheta_O)) / v))) / v) / ((v * 2.0f) * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) + 1.0f) / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_o * (costheta_i - ((costheta_i * (sintheta_i * sintheta_o)) / v))) / v) / ((v * 2.0e0) * ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) + 1.0e0) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

10. Simplified 69.8%

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

11. Final simplification 69.8%

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

Alternative 11: 64.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o / ((v / (costheta_i - (costheta_i * ((sintheta_i * sintheta_o) / v)))) / ((0.5e0 / v) / (((0.16666666666666666e0 / (v * v)) + 1.0e0) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

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

   2. exp-neg N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f32 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

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

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

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

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

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.5%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}\right), \color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{1}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   11. un-div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \left(\frac{v}{sinTheta\_i \cdot sinTheta\_O}\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \left(sinTheta\_i \cdot sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{1}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   16. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{v}{\color{blue}{\frac{1}{2}}}\right)\right) \]

   17. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{2}}{v}}}\right)\right) \]

9. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}} \]
10. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{cosTheta\_O}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}} \cdot \frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}} \]

    2. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{1}{2}}{v}} \cdot \frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}\right)}\right) \]

    3. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\frac{1}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}} \cdot \frac{\color{blue}{v}}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}\right)\right) \]

    4. associate-*l/ N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\frac{1 \cdot \frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}{\color{blue}{\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}}}\right)\right) \]

    5. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\frac{1 \cdot \frac{1}{\frac{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}}}{\frac{\frac{\frac{1}{2}}{\color{blue}{v}}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \]

    6. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\frac{\frac{1}{\frac{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}}}{\frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \]

    7. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\frac{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}{\frac{\color{blue}{\frac{\frac{1}{2}}{v}}}{\sinh \left(\frac{1}{v}\right)}}\right)\right) \]

    8. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}\right), \color{blue}{\left(\frac{\frac{\frac{1}{2}}{v}}{\sinh \left(\frac{1}{v}\right)}\right)}\right)\right) \]

11. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\frac{cosTheta\_O}{\frac{\frac{v}{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}}}} \]

12. Taylor expanded in v around inf

    \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}\right)\right)\right) \]
13. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), \color{blue}{v}\right)\right)\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right)\right)\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right)\right)\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right)\right)\right)\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right)\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right)\right)\right)\right) \]

    7. *-lowering-*.f32 63.4%

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right)\right)\right)\right) \]

14. Simplified 63.4%

    \[\leadsto \frac{cosTheta\_O}{\frac{\frac{v}{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{\frac{\frac{0.5}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}}}}} \]

15. Final simplification 63.4%

    \[\leadsto \frac{cosTheta\_O}{\frac{\frac{v}{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\frac{\frac{0.5}{v}}{\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v}}}} \]
16. Add Preprocessing

Alternative 12: 64.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_O}{\frac{2 + \frac{0.3333333333333333}{v \cdot v}}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  cosTheta_O
  (/
   (+ 2.0 (/ 0.3333333333333333 (* v v)))
   (/ (- cosTheta_i (* cosTheta_i (/ (* sinTheta_i sinTheta_O) v))) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O / ((2.0f + (0.3333333333333333f / (v * v))) / ((cosTheta_i - (cosTheta_i * ((sinTheta_i * sinTheta_O) / v))) / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o / ((2.0e0 + (0.3333333333333333e0 / (v * v))) / ((costheta_i - (costheta_i * ((sintheta_i * sintheta_o) / v))) / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O / Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))) / Float32(Float32(cosTheta_i - Float32(cosTheta_i * Float32(Float32(sinTheta_i * sinTheta_O) / v))) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O / ((single(2.0) + (single(0.3333333333333333) / (v * v))) / ((cosTheta_i - (cosTheta_i * ((sinTheta_i * sinTheta_O) / v))) / v));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.5%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}\right), \color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{1}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   11. un-div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \left(\frac{v}{sinTheta\_i \cdot sinTheta\_O}\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \left(sinTheta\_i \cdot sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{1}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   16. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{v}{\color{blue}{\frac{1}{2}}}\right)\right) \]

   17. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{2}}{v}}}\right)\right) \]

9. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}} \]

10. Taylor expanded in v around inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{\color{blue}{v}}^{2}}\right)\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right) \]

    6. *-lowering-*.f32 63.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right) \]

12. Simplified 63.3%

    \[\leadsto \frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]
13. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \frac{cosTheta\_O}{\color{blue}{\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) \cdot \frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}} \]

    2. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \color{blue}{\left(\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) \cdot \frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}\right)}\right) \]

    3. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) \cdot \frac{1}{\color{blue}{\frac{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}}}\right)\right) \]

    4. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) \cdot \frac{1}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{v}}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) \cdot \frac{1}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}}{v}}\right)\right) \]

    6. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right) \cdot \frac{1}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}}\right)\right) \]

    7. un-div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \left(\frac{2 + \frac{\frac{1}{3}}{v \cdot v}}{\color{blue}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}}}\right)\right) \]

    8. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(2 + \frac{\frac{1}{3}}{v \cdot v}\right), \color{blue}{\left(\frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}\right)}\right)\right) \]

    9. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{v \cdot v}\right)\right), \left(\frac{\color{blue}{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v}\right)\right)\right) \]

    10. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot v\right)\right)\right), \left(\frac{cosTheta\_i - \color{blue}{cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v}\right)\right)\right) \]

    11. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \left(\frac{cosTheta\_i - cosTheta\_i \cdot \color{blue}{\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{v}\right)\right)\right) \]

    12. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right), \mathsf{/.f32}\left(\left(cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), \color{blue}{v}\right)\right)\right) \]

14. Applied egg-rr 63.4%

    \[\leadsto \color{blue}{\frac{cosTheta\_O}{\frac{2 + \frac{0.3333333333333333}{v \cdot v}}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}}}} \]

15. Final simplification 63.4%

    \[\leadsto \frac{cosTheta\_O}{\frac{2 + \frac{0.3333333333333333}{v \cdot v}}{\frac{cosTheta\_i - cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}}{v}}} \]
16. Add Preprocessing

Alternative 13: 64.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ (* cosTheta_i cosTheta_O) v) (+ 2.0 (/ 0.3333333333333333 (* v v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * cosTheta_O) / v) / (2.0f + (0.3333333333333333f / (v * v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_i * costheta_o) / v) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * cosTheta_O) / v) / (single(2.0) + (single(0.3333333333333333) / (v * v)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.5%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}\right), \color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{1}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   11. un-div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \left(\frac{v}{sinTheta\_i \cdot sinTheta\_O}\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \left(sinTheta\_i \cdot sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{1}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   16. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{v}{\color{blue}{\frac{1}{2}}}\right)\right) \]

   17. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{2}}{v}}}\right)\right) \]

9. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}} \]

10. Taylor expanded in v around inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{\color{blue}{v}}^{2}}\right)\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right) \]

    6. *-lowering-*.f32 63.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right) \]

12. Simplified 63.3%

    \[\leadsto \frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

13. Taylor expanded in sinTheta_i around 0

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}} \]
14. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]

    2. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), \left(\color{blue}{2} + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)\right) \]

    5. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(2, \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) \]

    6. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{\color{blue}{v}}^{2}}\right)\right)\right) \]

    8. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right) \]

    10. *-lowering-*.f32 63.3%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right) \]

15. Simplified 63.3%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

16. Final simplification 63.3%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
17. Add Preprocessing

Alternative 14: 64.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i}}}{2 + \frac{0.3333333333333333}{v \cdot v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ cosTheta_O (/ v cosTheta_i)) (+ 2.0 (/ 0.3333333333333333 (* v v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_O / (v / cosTheta_i)) / (2.0f + (0.3333333333333333f / (v * v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_o / (v / costheta_i)) / (2.0e0 + (0.3333333333333333e0 / (v * v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O / Float32(v / cosTheta_i)) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O / (v / cosTheta_i)) / (single(2.0) + (single(0.3333333333333333) / (v * v)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v} + cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)}, \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + -1 \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i - cosTheta\_O \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, v\right)}\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{v}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(sinTheta\_O \cdot sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

   11. *-lowering-*.f32 98.5%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right), v\right)\right)\right), v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, 2\right)\right)\right) \]

7. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O \cdot \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}{v}\right), \color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot \frac{1}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}}\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(\frac{v}{cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right)\right), \left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot \left(v \cdot 2\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \left(cosTheta\_i - \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   7. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(cosTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   11. un-div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \left(\frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \left(\frac{v}{sinTheta\_i \cdot sinTheta\_O}\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \left(sinTheta\_i \cdot sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{1}{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

   16. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{v}{\color{blue}{\frac{1}{2}}}\right)\right) \]

   17. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{2}}{v}}}\right)\right) \]

9. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}}} \]

10. Taylor expanded in v around inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \color{blue}{\left(2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{v}^{2}}\right)}\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3} \cdot 1}{\color{blue}{{v}^{2}}}\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{1}{3}}{{\color{blue}{v}}^{2}}\right)\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \color{blue}{\left({v}^{2}\right)}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \left(v \cdot \color{blue}{v}\right)\right)\right)\right) \]

    6. *-lowering-*.f32 63.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, \mathsf{\_.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right) \]

12. Simplified 63.3%

    \[\leadsto \frac{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i - \frac{cosTheta\_i}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

13. Taylor expanded in v around inf

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{v}{cosTheta\_i}\right)}\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right) \]
14. Step-by-step derivation

    1. /-lowering-/.f32 63.3%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(v, cosTheta\_i\right)\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{1}{3}, \mathsf{*.f32}\left(v, v\right)\right)\right)\right) \]

15. Simplified 63.3%

    \[\leadsto \frac{\frac{cosTheta\_O}{\color{blue}{\frac{v}{cosTheta\_i}}}}{2 + \frac{0.3333333333333333}{v \cdot v}} \]
16. Add Preprocessing

Alternative 15: 59.2% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{v}}{\frac{1}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (/ 0.5 v) (/ 1.0 (* cosTheta_i cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) / (1.0f / (cosTheta_i * cosTheta_O));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) / (1.0e0 / (costheta_i * costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) / Float32(Float32(1.0) / Float32(cosTheta_i * cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) / (single(1.0) / (cosTheta_i * cosTheta_O));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing
5. 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. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\left(v \cdot 2\right) \cdot \sinh \left(\frac{1}{v}\right)}{\frac{\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{1}{\left(v \cdot 2\right) \cdot \color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{v \cdot 2}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}}} \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{v \cdot 2}\right), \color{blue}{\left(\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2 \cdot v}\right), \left(\frac{\sinh \left(\frac{1}{v}\right)}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}\right)\right) \]

   7. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2}\right), v\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{\sinh \color{blue}{\left(\frac{1}{v}\right)}}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}\right)\right) \]

   10. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\frac{1}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}}\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(\frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\frac{\color{blue}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(\frac{e^{\color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right), \color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)}\right)\right)\right) \]

6. Applied egg-rr 93.1%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \frac{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}}} \]

7. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(\frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)}\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)\right) \]

   2. *-lowering-*.f32 58.0%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{cosTheta\_i}\right)\right)\right) \]

9. Simplified 58.0%

   \[\leadsto \frac{\frac{0.5}{v}}{\color{blue}{\frac{1}{cosTheta\_O \cdot cosTheta\_i}}} \]

10. Final simplification 58.0%

    \[\leadsto \frac{\frac{0.5}{v}}{\frac{1}{cosTheta\_i \cdot cosTheta\_O}} \]
11. Add Preprocessing

Alternative 16: 59.1% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ v (* 0.5 (* cosTheta_i cosTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / (0.5f * (cosTheta_i * cosTheta_O)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0 / (v / (0.5e0 * (costheta_i * costheta_o)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(v / Float32(Float32(0.5) * Float32(cosTheta_i * cosTheta_O))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / (single(0.5) * (cosTheta_i * cosTheta_O)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 57.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
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. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{1}{2}\right)\right)}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \left(\frac{1}{2} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)\right)\right) \]

   7. *-lowering-*.f32 58.0%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{cosTheta\_O}\right)\right)\right)\right) \]

9. Applied egg-rr 58.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{v}{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}}} \]
10. Add Preprocessing

Alternative 17: 58.7% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (* cosTheta_i cosTheta_O)) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * (cosTheta_i * cosTheta_O)) / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 * (costheta_i * costheta_o)) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * Float32(cosTheta_i * cosTheta_O)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * (cosTheta_i * cosTheta_O)) / v;
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. 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 57.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}\right), v\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \frac{1}{2}\right), v\right) \]

   3. *-lowering-*.f32 57.5%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \frac{1}{2}\right), v\right) \]

9. Applied egg-rr 57.5%

   \[\leadsto \frac{\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}}{v} \]

10. Final simplification 57.5%

    \[\leadsto \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \]
11. Add Preprocessing

Alternative 18: 58.7% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (* cosTheta_i cosTheta_O)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * (cosTheta_i * cosTheta_O);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * (costheta_i * costheta_o)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * Float32(cosTheta_i * cosTheta_O))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (cosTheta_i * cosTheta_O);
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 57.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{2}}{v} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

   5. /-lowering-/.f32 57.4%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 57.4%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{0.5}{v}} \]

10. Final simplification 57.4%

    \[\leadsto \frac{0.5}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]
11. Add Preprocessing

Alternative 19: 58.7% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \frac{cosTheta\_i}{v} \cdot \left(cosTheta\_O \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ cosTheta_i v) (* cosTheta_O 0.5)))

C

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

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

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

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 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \color{blue}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)}\right) \]

   2. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \cdot v\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\color{blue}{\sinh \left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \left(e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right)\right), \left(\left(\sinh \color{blue}{\left(\frac{1}{v}\right)} \cdot 2\right) \cdot v\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{2}\right) \cdot v\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_i \cdot sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(2 \cdot v\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{2} \cdot v\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(2 \cdot v\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(v \cdot \color{blue}{2}\right)\right)\right) \]

   16. *-lowering-*.f32 98.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, sinTheta\_O\right), v\right)\right)\right), \mathsf{*.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{*.f32}\left(v, \color{blue}{2}\right)\right)\right) \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{\color{blue}{v}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \color{blue}{v}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\left(\frac{1}{2} \cdot cosTheta\_O\right) \cdot cosTheta\_i\right), v\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_i \cdot \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(\frac{1}{2} \cdot cosTheta\_O\right)\right), v\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \left(cosTheta\_O \cdot \frac{1}{2}\right)\right), v\right) \]

   7. *-lowering-*.f32 57.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right)\right), v\right) \]

7. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)}{v}} \]
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 57.4%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \frac{1}{2}\right), \mathsf{/.f32}\left(cosTheta\_i, \color{blue}{v}\right)\right) \]

9. Applied egg-rr 57.4%

   \[\leadsto \color{blue}{\left(cosTheta\_O \cdot 0.5\right) \cdot \frac{cosTheta\_i}{v}} \]

10. Final simplification 57.4%

    \[\leadsto \frac{cosTheta\_i}{v} \cdot \left(cosTheta\_O \cdot 0.5\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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)))