HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.2%

Time: 14.0s

Alternatives: 15

Speedup: N/A×

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: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_i}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   3. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l/ 98.6%

      \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   7. associate-*r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   9. exp-neg 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   10. associate-/l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   11. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   12. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   13. associate-*l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   14. *-commutative 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   15. exp-prod 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

4. Taylor expanded in cosTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-/l* 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

   2. *-commutative 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

   3. *-commutative 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   4. rec-exp 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   5. unpow2 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   6. associate-*r/ 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

7. Taylor expanded in sinTheta_i around 0 98.7%

   \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
8. Step-by-step derivation

   1. associate-/l* 98.6%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

   2. rec-exp 98.6%

      \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

   3. associate-/r/ 98.7%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

   4. unpow2 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

   5. rec-exp 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

   6. distribute-neg-frac 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

   7. metadata-eval 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

9. Simplified 98.7%

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

    1. pow1 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]

    2. associate-/l* 98.7%

       \[\leadsto \frac{cosTheta_i}{{\left(\color{blue}{\frac{v}{\frac{cosTheta_O}{v}}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}} \]

11. Applied egg-rr 98.7%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]
12. Step-by-step derivation

    1. unpow1 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

    2. *-commutative 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}}} \]

13. Simplified 98.7%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}}} \]

14. Final simplification 98.7%

    \[\leadsto \frac{cosTheta_i}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}} \]

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   3. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l/ 98.6%

      \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   7. associate-*r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   9. exp-neg 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   10. associate-/l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   11. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   12. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   13. associate-*l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   14. *-commutative 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   15. exp-prod 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

4. Taylor expanded in sinTheta_O around 0 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{{v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
5. Step-by-step derivation

   1. times-frac 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{{v}^{2}} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

   2. unpow2 98.7%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{v \cdot v}} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

   3. rec-exp 98.7%

      \[\leadsto \frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}} \]

   4. distribute-neg-frac 98.7%

      \[\leadsto \frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

   5. metadata-eval 98.7%

      \[\leadsto \frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

7. Final simplification 98.7%

   \[\leadsto \frac{cosTheta_i}{v \cdot v} \cdot \frac{cosTheta_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]

Alternative 3: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_i}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \left(v \cdot \frac{v}{cosTheta_O}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   3. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l/ 98.6%

      \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   7. associate-*r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   9. exp-neg 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   10. associate-/l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   11. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   12. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   13. associate-*l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   14. *-commutative 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   15. exp-prod 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

4. Taylor expanded in cosTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-/l* 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

   2. *-commutative 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

   3. *-commutative 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   4. rec-exp 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   5. unpow2 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   6. associate-*r/ 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

7. Taylor expanded in sinTheta_i around 0 98.7%

   \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
8. Step-by-step derivation

   1. associate-/l* 98.6%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

   2. rec-exp 98.6%

      \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

   3. associate-/r/ 98.7%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

   4. unpow2 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

   5. rec-exp 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

   6. distribute-neg-frac 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

   7. metadata-eval 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

9. Simplified 98.7%

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

    1. pow1 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]

    2. associate-/l* 98.7%

       \[\leadsto \frac{cosTheta_i}{{\left(\color{blue}{\frac{v}{\frac{cosTheta_O}{v}}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}} \]

11. Applied egg-rr 98.7%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]
12. Step-by-step derivation

    1. unpow1 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

    2. *-commutative 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}}} \]

    3. associate-/r/ 98.6%

       \[\leadsto \frac{cosTheta_i}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \color{blue}{\left(\frac{v}{cosTheta_O} \cdot v\right)}} \]

13. Simplified 98.6%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \left(\frac{v}{cosTheta_O} \cdot v\right)}} \]

14. Final simplification 98.6%

    \[\leadsto \frac{cosTheta_i}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \left(v \cdot \frac{v}{cosTheta_O}\right)} \]

Alternative 4: 76.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} \mathbf{if}\;v \leq 0.33000001311302185:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v}{cosTheta_O} \cdot 2 + \left(0.3333333333333333 \cdot \frac{1}{v \cdot cosTheta_O} + 0.016666666666666666 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.33000001311302185)
   (/ cosTheta_i (* (/ (* v v) cosTheta_O) (+ (exp (/ 1.0 v)) -1.0)))
   (/
    cosTheta_i
    (+
     (* (/ v cosTheta_O) 2.0)
     (+
      (* 0.3333333333333333 (/ 1.0 (* v cosTheta_O)))
      (* 0.016666666666666666 (/ 1.0 (* cosTheta_O (pow v 3.0)))))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.33000001311302185f) {
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = cosTheta_i / (((v / cosTheta_O) * 2.0f) + ((0.3333333333333333f * (1.0f / (v * cosTheta_O))) + (0.016666666666666666f * (1.0f / (cosTheta_O * powf(v, 3.0f))))));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.33000001311302185e0) then
        tmp = costheta_i / (((v * v) / costheta_o) * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = costheta_i / (((v / costheta_o) * 2.0e0) + ((0.3333333333333333e0 * (1.0e0 / (v * costheta_o))) + (0.016666666666666666e0 * (1.0e0 / (costheta_o * (v ** 3.0e0))))))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.33000001311302185))
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v * v) / cosTheta_O) * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v / cosTheta_O) * Float32(2.0)) + Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / Float32(v * cosTheta_O))) + Float32(Float32(0.016666666666666666) * Float32(Float32(1.0) / Float32(cosTheta_O * (v ^ Float32(3.0))))))));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.33000001311302185))
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = cosTheta_i / (((v / cosTheta_O) * single(2.0)) + ((single(0.3333333333333333) * (single(1.0) / (v * cosTheta_O))) + (single(0.016666666666666666) * (single(1.0) / (cosTheta_O * (v ^ single(3.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.330000013

   1. Initial program 98.3%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.2%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.2%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.3%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.3%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.3%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.3%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.3%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.3%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.3%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.3%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.3%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.3%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.3%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.3%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.4%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 98.2%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.2%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.4%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.4%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 76.0%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{1}\right)} \]

   if 0.330000013 < v

   1. Initial program 98.7%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.7%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.9%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.9%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.9%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 99.0%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 76.8%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{2 \cdot \frac{v}{cosTheta_O} + \left(0.3333333333333333 \cdot \frac{1}{v \cdot cosTheta_O} + 0.016666666666666666 \cdot \frac{1}{{v}^{3} \cdot cosTheta_O}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.33000001311302185:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v}{cosTheta_O} \cdot 2 + \left(0.3333333333333333 \cdot \frac{1}{v \cdot cosTheta_O} + 0.016666666666666666 \cdot \frac{1}{cosTheta_O \cdot {v}^{3}}\right)}\\ \end{array} \]

Alternative 5: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5199999809265137:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.5199999809265137)
   (/ cosTheta_i (* (/ (* v v) cosTheta_O) (+ (exp (/ 1.0 v)) -1.0)))
   (*
    (* cosTheta_i (/ cosTheta_O v))
    (+
     0.5
     (-
      (/ 0.009722222222222222 (pow v 4.0))
      (/ 0.08333333333333333 (* v v)))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.5199999809265137f) {
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = (cosTheta_i * (cosTheta_O / v)) * (0.5f + ((0.009722222222222222f / powf(v, 4.0f)) - (0.08333333333333333f / (v * v))));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.5199999809265137e0) then
        tmp = costheta_i / (((v * v) / costheta_o) * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = (costheta_i * (costheta_o / v)) * (0.5e0 + ((0.009722222222222222e0 / (v ** 4.0e0)) - (0.08333333333333333e0 / (v * v))))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5199999809265137))
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v * v) / cosTheta_O) * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(0.5) + Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(4.0))) - Float32(Float32(0.08333333333333333) / Float32(v * v)))));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.5199999809265137))
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = (cosTheta_i * (cosTheta_O / v)) * (single(0.5) + ((single(0.009722222222222222) / (v ^ single(4.0))) - (single(0.08333333333333333) / (v * v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.519999981

   1. Initial program 98.4%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.3%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.3%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.5%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 98.3%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.3%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.5%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 71.7%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{1}\right)} \]

   if 0.519999981 < v

   1. Initial program 98.8%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.8%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 99.0%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \sinh \left(\frac{1}{v}\right)\right)\right)}} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \sinh \left(\frac{1}{v}\right)\right)\right)}} \]

   6. Taylor expanded in sinTheta_O around 0 98.9%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{1}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{1}{v}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

      2. rec-exp 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}} \]

      3. distribute-neg-frac 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

      4. metadata-eval 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

   8. Simplified 99.1%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

   9. Taylor expanded in v around inf 74.9%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\left(\left(0.5 + 0.009722222222222222 \cdot \frac{1}{{v}^{4}}\right) - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\left(0.5 + \left(0.009722222222222222 \cdot \frac{1}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right)} \]

       2. associate-*r/ 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\color{blue}{\frac{0.009722222222222222 \cdot 1}{{v}^{4}}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right) \]

       3. metadata-eval 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{\color{blue}{0.009722222222222222}}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right) \]

       4. unpow2 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{\color{blue}{v \cdot v}}\right)\right) \]

       5. associate-*r/ 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \color{blue}{\frac{0.08333333333333333 \cdot 1}{v \cdot v}}\right)\right) \]

       6. metadata-eval 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{\color{blue}{0.08333333333333333}}{v \cdot v}\right)\right) \]

   11. Simplified 74.9%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5199999809265137:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \]

Alternative 6: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} \mathbf{if}\;v \leq 0.5199999809265137:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{\frac{0.08333333333333333}{v}}{v}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.5199999809265137)
   (/ cosTheta_i (* (/ (* v v) cosTheta_O) (+ (exp (/ 1.0 v)) -1.0)))
   (*
    (* cosTheta_i (/ cosTheta_O v))
    (+
     0.5
     (-
      (/ 0.009722222222222222 (pow v 4.0))
      (/ (/ 0.08333333333333333 v) v))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.5199999809265137f) {
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = (cosTheta_i * (cosTheta_O / v)) * (0.5f + ((0.009722222222222222f / powf(v, 4.0f)) - ((0.08333333333333333f / v) / v)));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.5199999809265137e0) then
        tmp = costheta_i / (((v * v) / costheta_o) * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = (costheta_i * (costheta_o / v)) * (0.5e0 + ((0.009722222222222222e0 / (v ** 4.0e0)) - ((0.08333333333333333e0 / v) / v)))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.5199999809265137))
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v * v) / cosTheta_O) * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(0.5) + Float32(Float32(Float32(0.009722222222222222) / (v ^ Float32(4.0))) - Float32(Float32(Float32(0.08333333333333333) / v) / v))));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.5199999809265137))
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = (cosTheta_i * (cosTheta_O / v)) * (single(0.5) + ((single(0.009722222222222222) / (v ^ single(4.0))) - ((single(0.08333333333333333) / v) / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.519999981

   1. Initial program 98.4%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.3%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.3%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.4%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.5%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 98.3%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.3%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.5%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.5%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 71.7%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{1}\right)} \]

   if 0.519999981 < v

   1. Initial program 98.8%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.8%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 99.0%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 99.0%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 99.0%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \sinh \left(\frac{1}{v}\right)\right)\right)}} \]

   5. Applied egg-rr 98.9%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot \sinh \left(\frac{1}{v}\right)\right)\right)}} \]

   6. Taylor expanded in sinTheta_O around 0 98.9%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{1}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{1}{v}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

      2. rec-exp 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}} \]

      3. distribute-neg-frac 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

      4. metadata-eval 99.1%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

   8. Simplified 99.1%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{1}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

   9. Taylor expanded in v around inf 74.9%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\left(\left(0.5 + 0.009722222222222222 \cdot \frac{1}{{v}^{4}}\right) - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\left(0.5 + \left(0.009722222222222222 \cdot \frac{1}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right)} \]

       2. associate-*r/ 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\color{blue}{\frac{0.009722222222222222 \cdot 1}{{v}^{4}}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right) \]

       3. metadata-eval 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{\color{blue}{0.009722222222222222}}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{{v}^{2}}\right)\right) \]

       4. unpow2 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - 0.08333333333333333 \cdot \frac{1}{\color{blue}{v \cdot v}}\right)\right) \]

       5. associate-*r/ 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \color{blue}{\frac{0.08333333333333333 \cdot 1}{v \cdot v}}\right)\right) \]

       6. metadata-eval 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{\color{blue}{0.08333333333333333}}{v \cdot v}\right)\right) \]

       7. associate-/r* 74.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \color{blue}{\frac{\frac{0.08333333333333333}{v}}{v}}\right)\right) \]

   11. Simplified 74.9%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{\frac{0.08333333333333333}{v}}{v}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.5199999809265137:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{\frac{0.08333333333333333}{v}}{v}\right)\right)\\ \end{array} \]

Alternative 7: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\left(v \cdot \frac{v}{cosTheta_O}\right) \cdot \left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= v 0.46000000834465027)
   (/ cosTheta_i (* (/ (* v v) cosTheta_O) (+ (exp (/ 1.0 v)) -1.0)))
   (/
    cosTheta_i
    (*
     (* v (/ v cosTheta_O))
     (+ (/ 2.0 v) (/ 0.3333333333333333 (pow v 3.0)))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = cosTheta_i / ((v * (v / cosTheta_O)) * ((2.0f / v) + (0.3333333333333333f / powf(v, 3.0f))));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.46000000834465027e0) then
        tmp = costheta_i / (((v * v) / costheta_o) * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = costheta_i / ((v * (v / costheta_o)) * ((2.0e0 / v) + (0.3333333333333333e0 / (v ** 3.0e0))))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v * v) / cosTheta_O) * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(cosTheta_i / Float32(Float32(v * Float32(v / cosTheta_O)) * Float32(Float32(Float32(2.0) / v) + Float32(Float32(0.3333333333333333) / (v ^ Float32(3.0))))));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = cosTheta_i / ((v * (v / cosTheta_O)) * ((single(2.0) / v) + (single(0.3333333333333333) / (v ^ single(3.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   1. Initial program 98.5%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.4%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.6%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.6%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 73.6%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{1}\right)} \]

   if 0.460000008 < v

   1. Initial program 98.7%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.7%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.9%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.8%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.8%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.9%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.9%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 69.7%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \color{blue}{\left(2 \cdot \frac{1}{v} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\color{blue}{\frac{2 \cdot 1}{v}} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}\right)} \]

       2. metadata-eval 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\frac{\color{blue}{2}}{v} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}\right)} \]

       3. associate-*r/ 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\frac{2}{v} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{3}}}\right)} \]

       4. metadata-eval 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\frac{2}{v} + \frac{\color{blue}{0.3333333333333333}}{{v}^{3}}\right)} \]

   12. Simplified 69.7%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \color{blue}{\left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)}} \]
   13. Step-by-step derivation

       1. associate-/l* 69.7%

          \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v}{\frac{cosTheta_O}{v}}} \cdot \left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)} \]

       2. associate-/r/ 69.7%

          \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(\frac{v}{cosTheta_O} \cdot v\right)} \cdot \left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)} \]

   14. Applied egg-rr 69.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(\frac{v}{cosTheta_O} \cdot v\right)} \cdot \left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\left(v \cdot \frac{v}{cosTheta_O}\right) \cdot \left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)}\\ \end{array} \]

Alternative 8: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := \frac{v \cdot v}{cosTheta_O}\\ \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{t_0 \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{t_0 \cdot \left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* v v) cosTheta_O)))
   (if (<= v 0.46000000834465027)
     (/ cosTheta_i (* t_0 (+ (exp (/ 1.0 v)) -1.0)))
     (/ cosTheta_i (* t_0 (+ (/ 2.0 v) (/ 0.3333333333333333 (pow v 3.0))))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (v * v) / cosTheta_O;
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = cosTheta_i / (t_0 * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = cosTheta_i / (t_0 * ((2.0f / v) + (0.3333333333333333f / powf(v, 3.0f))));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (v * v) / costheta_o
    if (v <= 0.46000000834465027e0) then
        tmp = costheta_i / (t_0 * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = costheta_i / (t_0 * ((2.0e0 / v) + (0.3333333333333333e0 / (v ** 3.0e0))))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(v * v) / cosTheta_O)
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(cosTheta_i / Float32(t_0 * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(cosTheta_i / Float32(t_0 * Float32(Float32(Float32(2.0) / v) + Float32(Float32(0.3333333333333333) / (v ^ Float32(3.0))))));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (v * v) / cosTheta_O;
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = cosTheta_i / (t_0 * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = cosTheta_i / (t_0 * ((single(2.0) / v) + (single(0.3333333333333333) / (v ^ single(3.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   1. Initial program 98.5%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.4%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.6%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.6%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 73.6%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{1}\right)} \]

   if 0.460000008 < v

   1. Initial program 98.7%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.7%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.9%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.9%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.9%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.8%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.8%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.8%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.9%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 99.0%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 99.0%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.9%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.9%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.9%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 69.7%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \color{blue}{\left(2 \cdot \frac{1}{v} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\color{blue}{\frac{2 \cdot 1}{v}} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}\right)} \]

       2. metadata-eval 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\frac{\color{blue}{2}}{v} + 0.3333333333333333 \cdot \frac{1}{{v}^{3}}\right)} \]

       3. associate-*r/ 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\frac{2}{v} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{3}}}\right)} \]

       4. metadata-eval 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\frac{2}{v} + \frac{\color{blue}{0.3333333333333333}}{{v}^{3}}\right)} \]

   12. Simplified 69.7%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \color{blue}{\left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}\right)}\\ \end{array} \]

Alternative 9: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{v} + v \cdot 2\\ \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{t_0}{cosTheta_O}} - \frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ (/ 0.3333333333333333 v) (* v 2.0))))
   (if (<= v 0.46000000834465027)
     (/ cosTheta_i (* (/ v (/ cosTheta_O v)) (+ (exp (/ 1.0 v)) -1.0)))
     (-
      (/ cosTheta_i (/ t_0 cosTheta_O))
      (* (/ sinTheta_i v) (/ (* cosTheta_i (* cosTheta_O sinTheta_O)) t_0))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (0.3333333333333333f / v) + (v * 2.0f);
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = cosTheta_i / ((v / (cosTheta_O / v)) * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (cosTheta_O * sinTheta_O)) / t_0));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (0.3333333333333333e0 / v) + (v * 2.0e0)
    if (v <= 0.46000000834465027e0) then
        tmp = costheta_i / ((v / (costheta_o / v)) * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = (costheta_i / (t_0 / costheta_o)) - ((sintheta_i / v) * ((costheta_i * (costheta_o * sintheta_o)) / t_0))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(0.3333333333333333) / v) + Float32(v * Float32(2.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(cosTheta_i / Float32(Float32(v / Float32(cosTheta_O / v)) * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(Float32(cosTheta_i / Float32(t_0 / cosTheta_O)) - Float32(Float32(sinTheta_i / v) * Float32(Float32(cosTheta_i * Float32(cosTheta_O * sinTheta_O)) / t_0)));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (single(0.3333333333333333) / v) + (v * single(2.0));
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = cosTheta_i / ((v / (cosTheta_O / v)) * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (cosTheta_O * sinTheta_O)) / t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   1. Initial program 98.5%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.4%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.6%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.6%

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

       1. pow1 98.6%

          \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]

       2. associate-/l* 98.6%

          \[\leadsto \frac{cosTheta_i}{{\left(\color{blue}{\frac{v}{\frac{cosTheta_O}{v}}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}} \]

   11. Applied egg-rr 98.6%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]
   12. Step-by-step derivation

       1. unpow1 98.6%

          \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

       2. *-commutative 98.6%

          \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}}} \]

   13. Simplified 98.6%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}}} \]

   14. Taylor expanded in v around inf 73.6%

       \[\leadsto \frac{cosTheta_i}{\left(e^{\frac{1}{v}} - \color{blue}{1}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}} \]

   if 0.460000008 < v

   1. Initial program 98.7%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. associate-*l/ 98.7%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]

      2. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

      3. exp-neg 98.8%

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

      4. associate-*l/ 98.8%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

      5. *-lft-identity 98.8%

         \[\leadsto \frac{\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

      6. associate-/l/ 98.8%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

      7. associate-*l* 98.8%

         \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      8. associate-*l* 98.8%

         \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      9. *-commutative 98.8%

         \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      10. *-commutative 98.8%

          \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]

      11. associate-*l/ 98.8%

          \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

   4. Taylor expanded in v around inf 69.7%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   5. Taylor expanded in sinTheta_O around 0 69.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 69.7%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}{cosTheta_O}}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      2. associate-*r/ 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{v}} + 2 \cdot v}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      3. metadata-eval 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{\color{blue}{0.3333333333333333}}{v} + 2 \cdot v}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      4. *-commutative 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + \color{blue}{v \cdot 2}}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      5. mul-1-neg 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \color{blue}{\left(-\frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}\right)} \]

      6. times-frac 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\color{blue}{\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}}\right) \]

      7. *-commutative 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \color{blue}{\left(cosTheta_O \cdot sinTheta_O\right)}}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}\right) \]

      8. associate-*r/ 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\color{blue}{\frac{0.3333333333333333 \cdot 1}{v}} + 2 \cdot v}\right) \]

      9. metadata-eval 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{\color{blue}{0.3333333333333333}}{v} + 2 \cdot v}\right) \]

      10. *-commutative 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + \color{blue}{v \cdot 2}}\right) \]

   7. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + v \cdot 2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} - \frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + v \cdot 2}\\ \end{array} \]

Alternative 10: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{v} + v \cdot 2\\ \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{t_0}{cosTheta_O}} - \frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ (/ 0.3333333333333333 v) (* v 2.0))))
   (if (<= v 0.46000000834465027)
     (/ cosTheta_i (* (/ (* v v) cosTheta_O) (+ (exp (/ 1.0 v)) -1.0)))
     (-
      (/ cosTheta_i (/ t_0 cosTheta_O))
      (* (/ sinTheta_i v) (/ (* cosTheta_i (* cosTheta_O sinTheta_O)) t_0))))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (0.3333333333333333f / v) + (v * 2.0f);
	float tmp;
	if (v <= 0.46000000834465027f) {
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (cosTheta_O * sinTheta_O)) / t_0));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (0.3333333333333333e0 / v) + (v * 2.0e0)
    if (v <= 0.46000000834465027e0) then
        tmp = costheta_i / (((v * v) / costheta_o) * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = (costheta_i / (t_0 / costheta_o)) - ((sintheta_i / v) * ((costheta_i * (costheta_o * sintheta_o)) / t_0))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(0.3333333333333333) / v) + Float32(v * Float32(2.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.46000000834465027))
		tmp = Float32(cosTheta_i / Float32(Float32(Float32(v * v) / cosTheta_O) * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(Float32(cosTheta_i / Float32(t_0 / cosTheta_O)) - Float32(Float32(sinTheta_i / v) * Float32(Float32(cosTheta_i * Float32(cosTheta_O * sinTheta_O)) / t_0)));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (single(0.3333333333333333) / v) + (v * single(2.0));
	tmp = single(0.0);
	if (v <= single(0.46000000834465027))
		tmp = cosTheta_i / (((v * v) / cosTheta_O) * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (cosTheta_O * sinTheta_O)) / t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.460000008

   1. Initial program 98.5%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      2. associate-*r/ 98.4%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

      3. *-commutative 98.4%

         \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      4. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      5. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

      6. *-commutative 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

      7. associate-*r* 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

      8. associate-/l/ 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

      9. exp-neg 98.4%

         \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      10. associate-/l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      11. associate-/r* 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      13. associate-*l/ 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      14. *-commutative 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

      15. exp-prod 98.4%

          \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   4. Taylor expanded in cosTheta_i around 0 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-/l* 98.6%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

      2. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

      3. *-commutative 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      4. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      5. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

      6. associate-*r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

   7. Taylor expanded in sinTheta_i around 0 98.5%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
   8. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

      2. rec-exp 98.4%

         \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

      3. associate-/r/ 98.6%

         \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

      4. unpow2 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

      5. rec-exp 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

      6. distribute-neg-frac 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

      7. metadata-eval 98.6%

         \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

   9. Simplified 98.6%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

   10. Taylor expanded in v around inf 73.6%

       \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{1}\right)} \]

   if 0.460000008 < v

   1. Initial program 98.7%

      \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
   2. Step-by-step derivation

      1. associate-*l/ 98.7%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]

      2. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

      3. exp-neg 98.8%

         \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

      4. associate-*l/ 98.8%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

      5. *-lft-identity 98.8%

         \[\leadsto \frac{\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

      6. associate-/l/ 98.8%

         \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

      7. associate-*l* 98.8%

         \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      8. associate-*l* 98.8%

         \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      9. *-commutative 98.8%

         \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

      10. *-commutative 98.8%

          \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]

      11. associate-*l/ 98.8%

          \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

   4. Taylor expanded in v around inf 69.7%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   5. Taylor expanded in sinTheta_O around 0 69.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 69.7%

         \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}{cosTheta_O}}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      2. associate-*r/ 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{v}} + 2 \cdot v}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      3. metadata-eval 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{\color{blue}{0.3333333333333333}}{v} + 2 \cdot v}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      4. *-commutative 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + \color{blue}{v \cdot 2}}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

      5. mul-1-neg 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \color{blue}{\left(-\frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}\right)} \]

      6. times-frac 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\color{blue}{\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}}\right) \]

      7. *-commutative 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \color{blue}{\left(cosTheta_O \cdot sinTheta_O\right)}}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}\right) \]

      8. associate-*r/ 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\color{blue}{\frac{0.3333333333333333 \cdot 1}{v}} + 2 \cdot v}\right) \]

      9. metadata-eval 69.7%

         \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{\color{blue}{0.3333333333333333}}{v} + 2 \cdot v}\right) \]

      10. *-commutative 69.7%

          \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + \color{blue}{v \cdot 2}}\right) \]

   7. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + v \cdot 2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.46000000834465027:\\ \;\;\;\;\frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} - \frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + v \cdot 2}\\ \end{array} \]

Alternative 11: 64.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{v} + v \cdot 2\\ \frac{cosTheta_i}{\frac{t_0}{cosTheta_O}} - \frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{t_0} \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ (/ 0.3333333333333333 v) (* v 2.0))))
   (-
    (/ cosTheta_i (/ t_0 cosTheta_O))
    (* (/ sinTheta_i v) (/ (* cosTheta_i (* cosTheta_O sinTheta_O)) t_0)))))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (0.3333333333333333f / v) + (v * 2.0f);
	return (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (cosTheta_O * sinTheta_O)) / t_0));
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = (0.3333333333333333e0 / v) + (v * 2.0e0)
    code = (costheta_i / (t_0 / costheta_o)) - ((sintheta_i / v) * ((costheta_i * (costheta_o * sintheta_o)) / t_0))
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(0.3333333333333333) / v) + Float32(v * Float32(2.0)))
	return Float32(Float32(cosTheta_i / Float32(t_0 / cosTheta_O)) - Float32(Float32(sinTheta_i / v) * Float32(Float32(cosTheta_i * Float32(cosTheta_O * sinTheta_O)) / t_0)))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (single(0.3333333333333333) / v) + (v * single(2.0));
	tmp = (cosTheta_i / (t_0 / cosTheta_O)) - ((sinTheta_i / v) * ((cosTheta_i * (cosTheta_O * sinTheta_O)) / t_0));
end

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. associate-*l/ 98.5%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]

   2. times-frac 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   3. exp-neg 98.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   4. associate-*l/ 98.7%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   5. *-lft-identity 98.7%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   6. associate-/l/ 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   7. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   8. associate-*l* 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   9. *-commutative 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   10. *-commutative 98.6%

       \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]

   11. associate-*l/ 98.6%

       \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

4. Taylor expanded in v around inf 64.8%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

5. Taylor expanded in sinTheta_O around 0 64.8%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}} \]
6. Step-by-step derivation

   1. associate-/l* 64.8%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}{cosTheta_O}}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   2. associate-*r/ 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{v}} + 2 \cdot v}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   3. metadata-eval 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{\color{blue}{0.3333333333333333}}{v} + 2 \cdot v}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   4. *-commutative 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + \color{blue}{v \cdot 2}}{cosTheta_O}} + -1 \cdot \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \]

   5. mul-1-neg 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \color{blue}{\left(-\frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)\right)}{v \cdot \left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)}\right)} \]

   6. times-frac 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\color{blue}{\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot cosTheta_O\right)}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}}\right) \]

   7. *-commutative 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \color{blue}{\left(cosTheta_O \cdot sinTheta_O\right)}}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}\right) \]

   8. associate-*r/ 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\color{blue}{\frac{0.3333333333333333 \cdot 1}{v}} + 2 \cdot v}\right) \]

   9. metadata-eval 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{\color{blue}{0.3333333333333333}}{v} + 2 \cdot v}\right) \]

   10. *-commutative 64.8%

       \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + \color{blue}{v \cdot 2}}\right) \]

7. Simplified 64.8%

   \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} + \left(-\frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + v \cdot 2}\right)} \]

8. Final simplification 64.8%

   \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} - \frac{sinTheta_i}{v} \cdot \frac{cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)}{\frac{0.3333333333333333}{v} + v \cdot 2} \]

Alternative 12: 64.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ cosTheta_i (/ (+ (/ 0.3333333333333333 v) (* v 2.0)) cosTheta_O)))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i / (((0.3333333333333333f / v) + (v * 2.0f)) / cosTheta_O);
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i / (((0.3333333333333333e0 / v) + (v * 2.0e0)) / costheta_o)
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i / (((single(0.3333333333333333) / v) + (v * single(2.0))) / cosTheta_O);
end

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. associate-*l/ 98.5%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]

   2. times-frac 98.7%

      \[\leadsto \color{blue}{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

   3. exp-neg 98.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   4. associate-*l/ 98.7%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   5. *-lft-identity 98.7%

      \[\leadsto \frac{\frac{\color{blue}{cosTheta_i \cdot cosTheta_O}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v} \]

   6. associate-/l/ 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

   7. associate-*l* 98.7%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot v\right)\right)} \cdot v\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   8. associate-*l* 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(2 \cdot v\right) \cdot v\right)\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   9. *-commutative 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\color{blue}{\left(v \cdot 2\right)} \cdot v\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

   10. *-commutative 98.6%

       \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{\color{blue}{sinTheta_O \cdot sinTheta_i}}{v}}} \]

   11. associate-*l/ 98.6%

       \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}} \]

4. Taylor expanded in v around inf 64.8%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

5. Taylor expanded in sinTheta_O around 0 64.8%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}} \]
6. Step-by-step derivation

   1. associate-/l* 64.8%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{0.3333333333333333 \cdot \frac{1}{v} + 2 \cdot v}{cosTheta_O}}} \]

   2. associate-*r/ 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\frac{0.3333333333333333 \cdot 1}{v}} + 2 \cdot v}{cosTheta_O}} \]

   3. metadata-eval 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{\color{blue}{0.3333333333333333}}{v} + 2 \cdot v}{cosTheta_O}} \]

   4. *-commutative 64.8%

      \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + \color{blue}{v \cdot 2}}{cosTheta_O}} \]

7. Simplified 64.8%

   \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}}} \]

8. Final simplification 64.8%

   \[\leadsto \frac{cosTheta_i}{\frac{\frac{0.3333333333333333}{v} + v \cdot 2}{cosTheta_O}} \]

Alternative 13: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   3. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l/ 98.6%

      \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   7. associate-*r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   9. exp-neg 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   10. associate-/l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   11. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   12. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   13. associate-*l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   14. *-commutative 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   15. exp-prod 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

4. Taylor expanded in v around inf 59.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
5. Step-by-step derivation

   1. associate-*l/ 59.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)} \]

6. Applied egg-rr 59.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O\right)} \]

7. Final simplification 59.2%

   \[\leadsto 0.5 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \]

Alternative 14: 58.5% accurate, 31.4× speedup

Math

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

FPCore

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

C

assert(cosTheta_i < cosTheta_O);
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

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

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
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

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
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 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   3. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l/ 98.6%

      \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   7. associate-*r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   9. exp-neg 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   10. associate-/l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   11. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   12. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   13. associate-*l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   14. *-commutative 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   15. exp-prod 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

4. Taylor expanded in v around inf 59.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]
5. Step-by-step derivation

   1. associate-*r/ 59.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v}} \]

   2. *-commutative 59.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(cosTheta_O \cdot cosTheta_i\right)}}{v} \]

6. Applied egg-rr 59.2%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta_O \cdot cosTheta_i\right)}{v}} \]

7. Final simplification 59.2%

   \[\leadsto \frac{0.5 \cdot \left(cosTheta_i \cdot cosTheta_O\right)}{v} \]

Alternative 15: 3.4% accurate, 73.3× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_i}{0} \end{array} \]

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ cosTheta_i 0.0))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i / 0.0f;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i / 0.0e0
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i / Float32(0.0))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i / single(0.0);
end

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. associate-*r/ 98.5%

      \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   3. *-commutative 98.5%

      \[\leadsto \frac{\color{blue}{cosTheta_O \cdot cosTheta_i}}{v} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-*l/ 98.6%

      \[\leadsto \color{blue}{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. *-commutative 98.6%

      \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. *-commutative 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}} \]

   7. associate-*r* 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{\color{blue}{\left(v \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot 2}} \]

   8. associate-/l/ 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \color{blue}{\frac{\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

   9. exp-neg 98.6%

      \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{\frac{1}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{2}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   10. associate-/l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{1}{2 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   11. associate-/r* 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\color{blue}{\frac{\frac{1}{2}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   12. metadata-eval 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{\color{blue}{0.5}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   13. associate-*l/ 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   14. *-commutative 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{e^{\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

   15. exp-prod 98.6%

       \[\leadsto \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{\color{blue}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}}{v \cdot \sinh \left(\frac{1}{v}\right)} \]

3. Simplified 98.6%

   \[\leadsto \color{blue}{\left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right) \cdot \frac{\frac{0.5}{{\left(e^{sinTheta_O}\right)}^{\left(\frac{sinTheta_i}{v}\right)}}}{v \cdot \sinh \left(\frac{1}{v}\right)}} \]

4. Taylor expanded in cosTheta_i around 0 98.6%

   \[\leadsto \color{blue}{\frac{cosTheta_i \cdot cosTheta_O}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}} \]
5. Step-by-step derivation

   1. associate-/l* 98.7%

      \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right)}{cosTheta_O}}} \]

   2. *-commutative 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left({v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}}{cosTheta_O}} \]

   3. *-commutative 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{\left(\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot {v}^{2}\right)} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   4. rec-exp 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right) \cdot {v}^{2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   5. unpow2 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}{cosTheta_O}} \]

   6. associate-*r/ 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{\color{blue}{sinTheta_i \cdot \frac{sinTheta_O}{v}}}}{cosTheta_O}} \]

6. Simplified 98.7%

   \[\leadsto \color{blue}{\frac{cosTheta_i}{\frac{\left(\left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right) \cdot \left(v \cdot v\right)\right) \cdot e^{sinTheta_i \cdot \frac{sinTheta_O}{v}}}{cosTheta_O}}} \]

7. Taylor expanded in sinTheta_i around 0 98.7%

   \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2} \cdot \left(e^{\frac{1}{v}} - e^{-\frac{1}{v}}\right)}{cosTheta_O}}} \]
8. Step-by-step derivation

   1. associate-/l* 98.6%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}}}}} \]

   2. rec-exp 98.6%

      \[\leadsto \frac{cosTheta_i}{\frac{{v}^{2}}{\frac{cosTheta_O}{e^{\frac{1}{v}} - \color{blue}{\frac{1}{e^{\frac{1}{v}}}}}}} \]

   3. associate-/r/ 98.7%

      \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]

   4. unpow2 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{\color{blue}{v \cdot v}}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

   5. rec-exp 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - \color{blue}{e^{-\frac{1}{v}}}\right)} \]

   6. distribute-neg-frac 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right)} \]

   7. metadata-eval 98.7%

      \[\leadsto \frac{cosTheta_i}{\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right)} \]

9. Simplified 98.7%

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

    1. pow1 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v \cdot v}{cosTheta_O} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]

    2. associate-/l* 98.7%

       \[\leadsto \frac{cosTheta_i}{{\left(\color{blue}{\frac{v}{\frac{cosTheta_O}{v}}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}} \]

11. Applied egg-rr 98.7%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{{\left(\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)\right)}^{1}}} \]
12. Step-by-step derivation

    1. unpow1 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\frac{v}{\frac{cosTheta_O}{v}} \cdot \left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right)}} \]

    2. *-commutative 98.7%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}}} \]

13. Simplified 98.7%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}}} \]
14. Step-by-step derivation

    1. expm1-log1p-u 45.8%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)\right)}} \]

    2. expm1-udef 45.8%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{e^{\mathsf{log1p}\left(\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1}} \]

    3. add-sqr-sqrt 45.8%

       \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\color{blue}{\sqrt{\frac{1}{v}} \cdot \sqrt{\frac{1}{v}}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    4. sqrt-unprod 45.8%

       \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\color{blue}{\sqrt{\frac{1}{v} \cdot \frac{1}{v}}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    5. frac-times 45.8%

       \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\sqrt{\color{blue}{\frac{1 \cdot 1}{v \cdot v}}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    6. metadata-eval 45.8%

       \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\sqrt{\frac{\color{blue}{1}}{v \cdot v}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    7. metadata-eval 45.8%

       \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\sqrt{\frac{\color{blue}{-1 \cdot -1}}{v \cdot v}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    8. frac-times 45.8%

       \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\sqrt{\color{blue}{\frac{-1}{v} \cdot \frac{-1}{v}}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    9. sqrt-unprod -0.0%

       \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\color{blue}{\sqrt{\frac{-1}{v}} \cdot \sqrt{\frac{-1}{v}}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    10. add-sqr-sqrt 3.2%

        \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\color{blue}{\frac{-1}{v}}} - e^{\frac{-1}{v}}\right) \cdot \frac{v}{\frac{cosTheta_O}{v}}\right)} - 1} \]

    11. associate-/r/ 3.2%

        \[\leadsto \frac{cosTheta_i}{e^{\mathsf{log1p}\left(\left(e^{\frac{-1}{v}} - e^{\frac{-1}{v}}\right) \cdot \color{blue}{\left(\frac{v}{cosTheta_O} \cdot v\right)}\right)} - 1} \]

15. Applied egg-rr 3.2%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{e^{\mathsf{log1p}\left(\left(e^{\frac{-1}{v}} - e^{\frac{-1}{v}}\right) \cdot \left(\frac{v}{cosTheta_O} \cdot v\right)\right)} - 1}} \]
16. Step-by-step derivation

    1. expm1-def 3.5%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(e^{\frac{-1}{v}} - e^{\frac{-1}{v}}\right) \cdot \left(\frac{v}{cosTheta_O} \cdot v\right)\right)\right)}} \]

    2. expm1-log1p 3.5%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{\left(e^{\frac{-1}{v}} - e^{\frac{-1}{v}}\right) \cdot \left(\frac{v}{cosTheta_O} \cdot v\right)}} \]

    3. +-inverses 3.5%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{0} \cdot \left(\frac{v}{cosTheta_O} \cdot v\right)} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{cosTheta_i}{\color{blue}{0}} \]

17. Simplified 3.2%

    \[\leadsto \frac{cosTheta_i}{\color{blue}{0}} \]

18. Final simplification 3.2%

    \[\leadsto \frac{cosTheta_i}{0} \]

Reproduce

herbie shell --seed 2023240 
(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)))