HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 14.1s

Alternatives: 17

Speedup: 1.0×

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: 98.7% accurate, 0.7× speedup

Math

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

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 (* (sinh (/ 1.0 v)) (* v (* v 2.0))))
  (/ cosTheta_O (pow (exp (/ sinTheta_O v)) sinTheta_i))))

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 / (sinhf((1.0f / v)) * (v * (v * 2.0f)))) * (cosTheta_O / powf(expf((sinTheta_O / v)), sinTheta_i));
}

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 / (sinh((1.0e0 / v)) * (v * (v * 2.0e0)))) * (costheta_o / (exp((sintheta_o / v)) ** sintheta_i))
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(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0))))) * Float32(cosTheta_O / (exp(Float32(sinTheta_O / v)) ^ sinTheta_i)))
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 / (sinh((single(1.0) / v)) * (v * (v * single(2.0))))) * (cosTheta_O / (exp((sinTheta_O / v)) ^ sinTheta_i));
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)\right)} \]

   2. expm1-udef 53.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)} - 1} \]

   3. times-frac 53.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}\right)} - 1 \]

   4. *-commutative 53.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)}} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)} - 1 \]

   5. exp-prod 53.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \cdot \frac{cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}\right)} - 1 \]

5. Applied egg-rr 53.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \cdot \frac{cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \cdot \frac{cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}\right)\right)} \]

   2. expm1-log1p 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

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
 (/
  (*
   (exp (/ (* sinTheta_O (- sinTheta_i)) v))
   (* (/ 1.0 v) (* cosTheta_i cosTheta_O)))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

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 = (exp(((sintheta_o * -sintheta_i) / v)) * ((1.0e0 / v) * (costheta_i * costheta_o))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
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(Float32(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i * cosTheta_O))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
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 = (exp(((sinTheta_O * -sinTheta_i) / v)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

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. div-inv 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

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 cosTheta_O)
  (*
   (exp (* (/ sinTheta_O v) sinTheta_i))
   (* v (* 2.0 (* v (sinh (/ 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 * cosTheta_O) / (expf(((sinTheta_O / v) * sinTheta_i)) * (v * (2.0f * (v * sinhf((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 * costheta_o) / (exp(((sintheta_o / v) * sintheta_i)) * (v * (2.0e0 * (v * sinh((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 * cosTheta_O) / Float32(exp(Float32(Float32(sinTheta_O / v) * sinTheta_i)) * Float32(v * Float32(Float32(2.0) * Float32(v * sinh(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 * cosTheta_O) / (exp(((sinTheta_O / v) * sinTheta_i)) * (v * (single(2.0) * (v * sinh((single(1.0) / v))))));
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   2. expm1-udef 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(e^{\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)} - 1\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   3. *-commutative 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\left(e^{\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)}\right)} - 1\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

5. Applied egg-rr 98.6%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\left(e^{\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)} - 1\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]
6. Step-by-step derivation

   1. expm1-def 98.6%

      \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)\right)} \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}} \]

   2. expm1-log1p 98.8%

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

   3. *-commutative 98.8%

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

   4. associate-*l* 98.7%

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

   5. *-commutative 98.7%

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

   6. associate-*l* 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

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 cosTheta_O)
  (*
   (* (sinh (/ 1.0 v)) (* v (* v 2.0)))
   (exp (* (/ sinTheta_O v) sinTheta_i)))))

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 * cosTheta_O) / ((sinhf((1.0f / v)) * (v * (v * 2.0f))) * expf(((sinTheta_O / v) * sinTheta_i)));
}

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 * costheta_o) / ((sinh((1.0e0 / v)) * (v * (v * 2.0e0))) * exp(((sintheta_o / v) * sintheta_i)))
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 * cosTheta_O) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0)))) * exp(Float32(Float32(sinTheta_O / v) * sinTheta_i))))
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 * cosTheta_O) / ((sinh((single(1.0) / v)) * (v * (v * single(2.0)))) * exp(((sinTheta_O / v) * sinTheta_i)));
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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. Final simplification 98.8%

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

Alternative 5: 98.5% 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 98.7%

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

   1. associate-*l/ 98.6%

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

   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* 98.8%

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

   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 sinTheta_O around 0 97.7%

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

   1. times-frac 97.7%

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

   2. unpow2 97.7%

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

   3. rec-exp 97.8%

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

   4. distribute-neg-frac 97.8%

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

   5. metadata-eval 97.8%

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

6. Simplified 97.8%

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

7. Final simplification 97.8%

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

Alternative 6: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \mathbf{if}\;v \leq 0.33000001311302185:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{v} \cdot \frac{1}{e^{\frac{1}{v}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{2 + \left(\frac{0.3333333333333333}{v \cdot v} + \frac{0.016666666666666666}{{v}^{4}}\right)}\\ \end{array} \end{array} \]

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 (* cosTheta_O (/ cosTheta_i v))))
   (if (<= v 0.33000001311302185)
     (* t_0 (* (/ 1.0 v) (/ 1.0 (+ (exp (/ 1.0 v)) -1.0))))
     (*
      t_0
      (/
       1.0
       (+
        2.0
        (+
         (/ 0.3333333333333333 (* v v))
         (/ 0.016666666666666666 (pow v 4.0)))))))))

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 = cosTheta_O * (cosTheta_i / v);
	float tmp;
	if (v <= 0.33000001311302185f) {
		tmp = t_0 * ((1.0f / v) * (1.0f / (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = t_0 * (1.0f / (2.0f + ((0.3333333333333333f / (v * v)) + (0.016666666666666666f / powf(v, 4.0f)))));
	}
	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 = costheta_o * (costheta_i / v)
    if (v <= 0.33000001311302185e0) then
        tmp = t_0 * ((1.0e0 / v) * (1.0e0 / (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = t_0 * (1.0e0 / (2.0e0 + ((0.3333333333333333e0 / (v * v)) + (0.016666666666666666e0 / (v ** 4.0e0)))))
    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(cosTheta_O * Float32(cosTheta_i / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.33000001311302185))
		tmp = Float32(t_0 * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(t_0 * Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(Float32(0.016666666666666666) / (v ^ Float32(4.0)))))));
	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 = cosTheta_O * (cosTheta_i / v);
	tmp = single(0.0);
	if (v <= single(0.33000001311302185))
		tmp = t_0 * ((single(1.0) / v) * (single(1.0) / (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = t_0 * (single(1.0) / (single(2.0) + ((single(0.3333333333333333) / (v * v)) + (single(0.016666666666666666) / (v ^ single(4.0))))));
	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. associate-/l* 98.2%

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

      4. associate-/r/ 98.1%

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

      5. *-commutative 98.1%

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

      6. *-commutative 98.1%

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

      7. associate-*r* 98.1%

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

      8. associate-/l/ 98.1%

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

      9. exp-neg 98.1%

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

      10. associate-/l/ 98.1%

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

      11. associate-/r* 98.1%

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

      12. metadata-eval 98.1%

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

      13. associate-*l/ 98.1%

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

      14. *-commutative 98.1%

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

      15. exp-prod 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in sinTheta_O around 0 96.8%

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

      1. rec-exp 96.8%

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

      2. distribute-neg-frac 96.8%

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

      3. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

      1. inv-pow 96.8%

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

      2. unpow-prod-down 97.1%

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

      3. inv-pow 97.1%

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

   8. Applied egg-rr 97.1%

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

      1. unpow-1 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in v around inf 74.6%

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

   if 0.330000013 < v

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r/ 99.0%

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

      3. associate-/l* 98.9%

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

      4. associate-/r/ 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

      8. associate-/l/ 98.9%

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

      9. exp-neg 98.9%

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

      10. associate-/l/ 98.9%

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

      11. associate-/r* 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-*l/ 98.9%

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

      14. *-commutative 98.9%

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

      15. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in sinTheta_O around 0 98.0%

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

      1. rec-exp 98.1%

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

      2. distribute-neg-frac 98.1%

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

      3. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in v around inf 77.2%

      \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{2 + \left(0.3333333333333333 \cdot \frac{1}{{v}^{2}} + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ 77.2%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{2 + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}} + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)} \]

      2. metadata-eval 77.2%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{2 + \left(\frac{\color{blue}{0.3333333333333333}}{{v}^{2}} + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)} \]

      3. unpow2 77.2%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{2 + \left(\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)} \]

      4. associate-*r/ 77.2%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{2 + \left(\frac{0.3333333333333333}{v \cdot v} + \color{blue}{\frac{0.016666666666666666 \cdot 1}{{v}^{4}}}\right)} \]

      5. metadata-eval 77.2%

         \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{2 + \left(\frac{0.3333333333333333}{v \cdot v} + \frac{\color{blue}{0.016666666666666666}}{{v}^{4}}\right)} \]

   9. Simplified 77.2%

      \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{2 + \left(\frac{0.3333333333333333}{v \cdot v} + \frac{0.016666666666666666}{{v}^{4}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.33000001311302185:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(\frac{1}{v} \cdot \frac{1}{e^{\frac{1}{v}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{2 + \left(\frac{0.3333333333333333}{v \cdot v} + \frac{0.016666666666666666}{{v}^{4}}\right)}\\ \end{array} \]

Alternative 7: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \mathbf{if}\;v \leq 0.33000001311302185:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{v} \cdot \frac{1}{e^{\frac{1}{v}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{\frac{0.3333333333333333}{v \cdot v} + \left(2 + \frac{0.016666666666666666}{{v}^{4}}\right)}\\ \end{array} \end{array} \]

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 (* cosTheta_O (/ cosTheta_i v))))
   (if (<= v 0.33000001311302185)
     (* t_0 (* (/ 1.0 v) (/ 1.0 (+ (exp (/ 1.0 v)) -1.0))))
     (*
      t_0
      (/
       1.0
       (+
        (/ 0.3333333333333333 (* v v))
        (+ 2.0 (/ 0.016666666666666666 (pow v 4.0)))))))))

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 = cosTheta_O * (cosTheta_i / v);
	float tmp;
	if (v <= 0.33000001311302185f) {
		tmp = t_0 * ((1.0f / v) * (1.0f / (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = t_0 * (1.0f / ((0.3333333333333333f / (v * v)) + (2.0f + (0.016666666666666666f / powf(v, 4.0f)))));
	}
	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 = costheta_o * (costheta_i / v)
    if (v <= 0.33000001311302185e0) then
        tmp = t_0 * ((1.0e0 / v) * (1.0e0 / (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = t_0 * (1.0e0 / ((0.3333333333333333e0 / (v * v)) + (2.0e0 + (0.016666666666666666e0 / (v ** 4.0e0)))))
    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(cosTheta_O * Float32(cosTheta_i / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.33000001311302185))
		tmp = Float32(t_0 * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(t_0 * Float32(Float32(1.0) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(Float32(2.0) + Float32(Float32(0.016666666666666666) / (v ^ Float32(4.0)))))));
	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 = cosTheta_O * (cosTheta_i / v);
	tmp = single(0.0);
	if (v <= single(0.33000001311302185))
		tmp = t_0 * ((single(1.0) / v) * (single(1.0) / (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = t_0 * (single(1.0) / ((single(0.3333333333333333) / (v * v)) + (single(2.0) + (single(0.016666666666666666) / (v ^ single(4.0))))));
	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. associate-/l* 98.2%

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

      4. associate-/r/ 98.1%

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

      5. *-commutative 98.1%

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

      6. *-commutative 98.1%

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

      7. associate-*r* 98.1%

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

      8. associate-/l/ 98.1%

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

      9. exp-neg 98.1%

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

      10. associate-/l/ 98.1%

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

      11. associate-/r* 98.1%

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

      12. metadata-eval 98.1%

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

      13. associate-*l/ 98.1%

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

      14. *-commutative 98.1%

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

      15. exp-prod 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in sinTheta_O around 0 96.8%

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

      1. rec-exp 96.8%

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

      2. distribute-neg-frac 96.8%

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

      3. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

      1. inv-pow 96.8%

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

      2. unpow-prod-down 97.1%

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

      3. inv-pow 97.1%

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

   8. Applied egg-rr 97.1%

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

      1. unpow-1 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in v around inf 74.6%

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

   if 0.330000013 < v

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*r/ 99.0%

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

      3. associate-/l* 98.9%

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

      4. associate-/r/ 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

      8. associate-/l/ 98.9%

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

      9. exp-neg 98.9%

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

      10. associate-/l/ 98.9%

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

      11. associate-/r* 98.9%

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

      12. metadata-eval 98.9%

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

      13. associate-*l/ 98.9%

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

      14. *-commutative 98.9%

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

      15. exp-prod 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in sinTheta_O around 0 98.0%

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

      1. rec-exp 98.1%

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

      2. distribute-neg-frac 98.1%

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

      3. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

      1. add-exp-log 97.9%

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

   8. Applied egg-rr 97.9%

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

   9. Taylor expanded in v around inf 77.2%

      \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{2 + \left(0.3333333333333333 \cdot \frac{1}{{v}^{2}} + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)}} \]
   10. Step-by-step derivation

       1. associate-+r+ 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{\left(2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}\right) + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}}} \]

       2. +-commutative 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{v}^{2}} + 2\right)} + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}} \]

       3. associate-+l+ 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{0.3333333333333333 \cdot \frac{1}{{v}^{2}} + \left(2 + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)}} \]

       4. associate-*r/ 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{\frac{0.3333333333333333 \cdot 1}{{v}^{2}}} + \left(2 + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)} \]

       5. metadata-eval 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\frac{\color{blue}{0.3333333333333333}}{{v}^{2}} + \left(2 + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)} \]

       6. unpow2 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + \left(2 + 0.016666666666666666 \cdot \frac{1}{{v}^{4}}\right)} \]

       7. associate-*r/ 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\frac{0.3333333333333333}{v \cdot v} + \left(2 + \color{blue}{\frac{0.016666666666666666 \cdot 1}{{v}^{4}}}\right)} \]

       8. metadata-eval 77.2%

          \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\frac{0.3333333333333333}{v \cdot v} + \left(2 + \frac{\color{blue}{0.016666666666666666}}{{v}^{4}}\right)} \]

   11. Simplified 77.2%

       \[\leadsto \left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + \left(2 + \frac{0.016666666666666666}{{v}^{4}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.33000001311302185:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(\frac{1}{v} \cdot \frac{1}{e^{\frac{1}{v}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{\frac{0.3333333333333333}{v \cdot v} + \left(2 + \frac{0.016666666666666666}{{v}^{4}}\right)}\\ \end{array} \]

Alternative 8: 73.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := cosTheta_O \cdot \frac{cosTheta_i}{v}\\ \mathbf{if}\;v \leq 0.5199999809265137:\\ \;\;\;\;t_0 \cdot \left(\frac{1}{v} \cdot \frac{1}{e^{\frac{1}{v}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.5 + \left(\frac{0.009722222222222222}{{v}^{4}} - \frac{0.08333333333333333}{v \cdot v}\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* cosTheta_O (/ cosTheta_i v))))
   (if (<= v 0.5199999809265137)
     (* t_0 (* (/ 1.0 v) (/ 1.0 (+ (exp (/ 1.0 v)) -1.0))))
     (*
      t_0
      (+
       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 t_0 = cosTheta_O * (cosTheta_i / v);
	float tmp;
	if (v <= 0.5199999809265137f) {
		tmp = t_0 * ((1.0f / v) * (1.0f / (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = t_0 * (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) :: t_0
    real(4) :: tmp
    t_0 = costheta_o * (costheta_i / v)
    if (v <= 0.5199999809265137e0) then
        tmp = t_0 * ((1.0e0 / v) * (1.0e0 / (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = t_0 * (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)
	t_0 = Float32(cosTheta_O * Float32(cosTheta_i / v))
	tmp = Float32(0.0)
	if (v <= Float32(0.5199999809265137))
		tmp = Float32(t_0 * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(t_0 * 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)
	t_0 = cosTheta_O * (cosTheta_i / v);
	tmp = single(0.0);
	if (v <= single(0.5199999809265137))
		tmp = t_0 * ((single(1.0) / v) * (single(1.0) / (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = t_0 * (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. associate-/l* 98.4%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 96.6%

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

      1. rec-exp 96.6%

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

      2. distribute-neg-frac 96.6%

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

      3. metadata-eval 96.6%

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

   6. Simplified 96.6%

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

      1. inv-pow 96.6%

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

      2. unpow-prod-down 96.8%

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

      3. inv-pow 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. unpow-1 96.8%

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

   10. Simplified 96.8%

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

   11. Taylor expanded in v around inf 69.3%

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

   if 0.519999981 < v

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 99.2%

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

      3. associate-/l* 99.0%

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

      4. associate-/r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. *-commutative 99.1%

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

      7. associate-*r* 99.1%

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

      8. associate-/l/ 99.1%

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

      9. exp-neg 99.1%

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

      10. associate-/l/ 99.1%

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

      11. associate-/r* 99.1%

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

      12. metadata-eval 99.1%

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

      13. associate-*l/ 99.1%

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

      14. *-commutative 99.1%

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

      15. exp-prod 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in sinTheta_O around 0 98.9%

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

      1. rec-exp 99.0%

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

      2. distribute-neg-frac 99.0%

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

      3. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in v around inf 77.8%

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

      1. associate--l+ 77.8%

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

      2. associate-*r/ 77.8%

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

      3. metadata-eval 77.8%

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

      4. associate-*r/ 77.8%

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

      5. metadata-eval 77.8%

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

      6. unpow2 77.8%

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

   9. Simplified 77.8%

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

4. Final simplification 72.7%

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

Alternative 9: 73.6% accurate, 1.8× speedup

Math

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

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_O (/ cosTheta_i v))
    (* (/ 1.0 v) (/ 1.0 (+ (exp (/ 1.0 v)) -1.0))))
   (*
    (/ cosTheta_i v)
    (*
     cosTheta_O
     (+
      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_O * (cosTheta_i / v)) * ((1.0f / v) * (1.0f / (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = (cosTheta_i / v) * (cosTheta_O * (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_o * (costheta_i / v)) * ((1.0e0 / v) * (1.0e0 / (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = (costheta_i / v) * (costheta_o * (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(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(cosTheta_i / v) * Float32(cosTheta_O * 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_O * (cosTheta_i / v)) * ((single(1.0) / v) * (single(1.0) / (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = (cosTheta_i / v) * (cosTheta_O * (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. associate-/l* 98.4%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 96.6%

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

      1. rec-exp 96.6%

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

      2. distribute-neg-frac 96.6%

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

      3. metadata-eval 96.6%

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

   6. Simplified 96.6%

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

      1. inv-pow 96.6%

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

      2. unpow-prod-down 96.8%

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

      3. inv-pow 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. unpow-1 96.8%

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

   10. Simplified 96.8%

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

   11. Taylor expanded in v around inf 69.3%

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

   if 0.519999981 < v

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 99.2%

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

      3. associate-/l* 99.0%

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

      4. associate-/r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. *-commutative 99.1%

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

      7. associate-*r* 99.1%

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

      8. associate-/l/ 99.1%

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

      9. exp-neg 99.1%

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

      10. associate-/l/ 99.1%

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

      11. associate-/r* 99.1%

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

      12. metadata-eval 99.1%

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

      13. associate-*l/ 99.1%

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

      14. *-commutative 99.1%

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

      15. exp-prod 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in sinTheta_O around 0 98.9%

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

      1. rec-exp 99.0%

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

      2. distribute-neg-frac 99.0%

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

      3. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in v around inf 77.8%

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

      1. associate--l+ 77.8%

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

      2. associate-*r/ 77.8%

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

      3. metadata-eval 77.8%

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

      4. associate-*r/ 77.8%

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

      5. metadata-eval 77.8%

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

      6. unpow2 77.8%

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

   9. Simplified 77.8%

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

   10. Taylor expanded in cosTheta_O around 0 77.8%

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

       1. associate-/l* 77.8%

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

       2. associate-/r/ 77.8%

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

       3. *-commutative 77.8%

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

       4. associate--l+ 77.8%

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

       5. sub-neg 77.8%

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

       6. associate-*r/ 77.8%

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

       7. metadata-eval 77.8%

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

       8. associate-*r/ 77.8%

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

       9. metadata-eval 77.8%

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

       10. distribute-neg-frac 77.8%

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

       11. metadata-eval 77.8%

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

       12. unpow2 77.8%

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

   12. Simplified 77.8%

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

4. Final simplification 72.7%

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

Alternative 10: 73.5% accurate, 1.8× speedup

Math

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

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_O (/ cosTheta_i v))
    (* (/ 1.0 v) (/ 1.0 (+ (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_O * (cosTheta_i / v)) * ((1.0f / v) * (1.0f / (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_o * (costheta_i / v)) * ((1.0e0 / v) * (1.0e0 / (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(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(Float32(cosTheta_i * 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_O * (cosTheta_i / v)) * ((single(1.0) / v) * (single(1.0) / (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. associate-/l* 98.4%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 96.6%

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

      1. rec-exp 96.6%

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

      2. distribute-neg-frac 96.6%

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

      3. metadata-eval 96.6%

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

   6. Simplified 96.6%

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

      1. inv-pow 96.6%

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

      2. unpow-prod-down 96.8%

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

      3. inv-pow 96.8%

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

   8. Applied egg-rr 96.8%

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

      1. unpow-1 96.8%

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

   10. Simplified 96.8%

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

   11. Taylor expanded in v around inf 69.3%

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

   if 0.519999981 < v

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*r/ 99.2%

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

      3. associate-/l* 99.0%

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

      4. associate-/r/ 99.1%

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

      5. *-commutative 99.1%

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

      6. *-commutative 99.1%

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

      7. associate-*r* 99.1%

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

      8. associate-/l/ 99.1%

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

      9. exp-neg 99.1%

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

      10. associate-/l/ 99.1%

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

      11. associate-/r* 99.1%

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

      12. metadata-eval 99.1%

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

      13. associate-*l/ 99.1%

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

      14. *-commutative 99.1%

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

      15. exp-prod 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in sinTheta_O around 0 98.9%

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

      1. rec-exp 99.0%

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

      2. distribute-neg-frac 99.0%

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

      3. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in v around inf 77.8%

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

      1. associate--l+ 77.8%

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

      2. associate-*r/ 77.8%

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

      3. metadata-eval 77.8%

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

      4. associate-*r/ 77.8%

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

      5. metadata-eval 77.8%

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

      6. unpow2 77.8%

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

   9. Simplified 77.8%

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

   10. Taylor expanded in cosTheta_O around 0 77.9%

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

4. Final simplification 72.7%

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

Alternative 11: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := \frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\\ \mathbf{if}\;v \leq 0.44600000977516174:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(\frac{1}{v} \cdot \frac{1}{e^{\frac{1}{v}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{t_0} - \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)\right)}{v \cdot t_0}\\ \end{array} \end{array} \]

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 (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))
   (if (<= v 0.44600000977516174)
     (*
      (* cosTheta_O (/ cosTheta_i v))
      (* (/ 1.0 v) (/ 1.0 (+ (exp (/ 1.0 v)) -1.0))))
     (-
      (/ (* cosTheta_i cosTheta_O) t_0)
      (/ (* sinTheta_i (* cosTheta_i (* cosTheta_O sinTheta_O))) (* v 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 = ((1.0f / v) * 0.3333333333333333f) + (v * 2.0f);
	float tmp;
	if (v <= 0.44600000977516174f) {
		tmp = (cosTheta_O * (cosTheta_i / v)) * ((1.0f / v) * (1.0f / (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = ((cosTheta_i * cosTheta_O) / t_0) - ((sinTheta_i * (cosTheta_i * (cosTheta_O * sinTheta_O))) / (v * 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 = ((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0)
    if (v <= 0.44600000977516174e0) then
        tmp = (costheta_o * (costheta_i / v)) * ((1.0e0 / v) * (1.0e0 / (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = ((costheta_i * costheta_o) / t_0) - ((sintheta_i * (costheta_i * (costheta_o * sintheta_o))) / (v * 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(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.44600000977516174))
		tmp = Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(Float32(cosTheta_i * cosTheta_O) / t_0) - Float32(Float32(sinTheta_i * Float32(cosTheta_i * Float32(cosTheta_O * sinTheta_O))) / Float32(v * 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(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0));
	tmp = single(0.0);
	if (v <= single(0.44600000977516174))
		tmp = (cosTheta_O * (cosTheta_i / v)) * ((single(1.0) / v) * (single(1.0) / (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = ((cosTheta_i * cosTheta_O) / t_0) - ((sinTheta_i * (cosTheta_i * (cosTheta_O * sinTheta_O))) / (v * t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.44600001

   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. associate-/l* 98.3%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 97.1%

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

      1. rec-exp 97.1%

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

      2. distribute-neg-frac 97.1%

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

      3. metadata-eval 97.1%

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

   6. Simplified 97.1%

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

      1. inv-pow 97.1%

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

      2. unpow-prod-down 97.3%

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

      3. inv-pow 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. unpow-1 97.3%

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

   10. Simplified 97.3%

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

   11. Taylor expanded in v around inf 71.1%

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

   if 0.44600001 < v

   1. Initial program 99.1%

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

      1. associate-*l/ 99.1%

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

      2. times-frac 99.1%

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

      3. exp-neg 99.1%

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

      4. associate-*l/ 99.1%

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

      5. *-lft-identity 99.1%

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

      6. associate-/l/ 99.1%

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

      7. associate-*l* 99.1%

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

      8. associate-*l* 99.3%

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

      9. *-commutative 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-*l/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in v around inf 72.4%

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

   5. Taylor expanded in sinTheta_O around 0 72.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44600000977516174:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \left(\frac{1}{v} \cdot \frac{1}{e^{\frac{1}{v}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} - \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)\right)}{v \cdot \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \]

Alternative 12: 72.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := \frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\\ \mathbf{if}\;v \leq 0.44600000977516174:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{t_0} - \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)\right)}{v \cdot t_0}\\ \end{array} \end{array} \]

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 (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))
   (if (<= v 0.44600000977516174)
     (* (* cosTheta_O (/ cosTheta_i v)) (/ 1.0 (* v (+ (exp (/ 1.0 v)) -1.0))))
     (-
      (/ (* cosTheta_i cosTheta_O) t_0)
      (/ (* sinTheta_i (* cosTheta_i (* cosTheta_O sinTheta_O))) (* v 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 = ((1.0f / v) * 0.3333333333333333f) + (v * 2.0f);
	float tmp;
	if (v <= 0.44600000977516174f) {
		tmp = (cosTheta_O * (cosTheta_i / v)) * (1.0f / (v * (expf((1.0f / v)) + -1.0f)));
	} else {
		tmp = ((cosTheta_i * cosTheta_O) / t_0) - ((sinTheta_i * (cosTheta_i * (cosTheta_O * sinTheta_O))) / (v * 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 = ((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0)
    if (v <= 0.44600000977516174e0) then
        tmp = (costheta_o * (costheta_i / v)) * (1.0e0 / (v * (exp((1.0e0 / v)) + (-1.0e0))))
    else
        tmp = ((costheta_i * costheta_o) / t_0) - ((sintheta_i * (costheta_i * (costheta_o * sintheta_o))) / (v * 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(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0)))
	tmp = Float32(0.0)
	if (v <= Float32(0.44600000977516174))
		tmp = Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) * Float32(Float32(1.0) / Float32(v * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(Float32(cosTheta_i * cosTheta_O) / t_0) - Float32(Float32(sinTheta_i * Float32(cosTheta_i * Float32(cosTheta_O * sinTheta_O))) / Float32(v * 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(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0));
	tmp = single(0.0);
	if (v <= single(0.44600000977516174))
		tmp = (cosTheta_O * (cosTheta_i / v)) * (single(1.0) / (v * (exp((single(1.0) / v)) + single(-1.0))));
	else
		tmp = ((cosTheta_i * cosTheta_O) / t_0) - ((sinTheta_i * (cosTheta_i * (cosTheta_O * sinTheta_O))) / (v * t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.44600001

   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. associate-/l* 98.3%

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

      4. associate-/r/ 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. associate-/l/ 98.2%

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

      9. exp-neg 98.2%

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

      10. associate-/l/ 98.2%

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

      11. associate-/r* 98.2%

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

      12. metadata-eval 98.2%

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

      13. associate-*l/ 98.2%

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

      14. *-commutative 98.2%

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

      15. exp-prod 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in sinTheta_O around 0 97.1%

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

      1. rec-exp 97.1%

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

      2. distribute-neg-frac 97.1%

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

      3. metadata-eval 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in v around inf 71.1%

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

   if 0.44600001 < v

   1. Initial program 99.1%

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

      1. associate-*l/ 99.1%

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

      2. times-frac 99.1%

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

      3. exp-neg 99.1%

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

      4. associate-*l/ 99.1%

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

      5. *-lft-identity 99.1%

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

      6. associate-/l/ 99.1%

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

      7. associate-*l* 99.1%

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

      8. associate-*l* 99.3%

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

      9. *-commutative 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-*l/ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in v around inf 72.4%

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

   5. Taylor expanded in sinTheta_O around 0 72.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.44600000977516174:\\ \;\;\;\;\left(cosTheta_O \cdot \frac{cosTheta_i}{v}\right) \cdot \frac{1}{v \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} - \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)\right)}{v \cdot \left(\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\right)}\\ \end{array} \]

Alternative 13: 65.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \begin{array}{l} t_0 := \frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2\\ \frac{cosTheta_i \cdot cosTheta_O}{t_0} - \frac{sinTheta_i \cdot \left(cosTheta_i \cdot \left(cosTheta_O \cdot sinTheta_O\right)\right)}{v \cdot t_0} \end{array} \end{array} \]

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 (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))
   (-
    (/ (* cosTheta_i cosTheta_O) t_0)
    (/ (* sinTheta_i (* cosTheta_i (* cosTheta_O sinTheta_O))) (* v 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 = ((1.0f / v) * 0.3333333333333333f) + (v * 2.0f);
	return ((cosTheta_i * cosTheta_O) / t_0) - ((sinTheta_i * (cosTheta_i * (cosTheta_O * sinTheta_O))) / (v * 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 = ((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0)
    code = ((costheta_i * costheta_o) / t_0) - ((sintheta_i * (costheta_i * (costheta_o * sintheta_o))) / (v * 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(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0)))
	return Float32(Float32(Float32(cosTheta_i * cosTheta_O) / t_0) - Float32(Float32(sinTheta_i * Float32(cosTheta_i * Float32(cosTheta_O * sinTheta_O))) / Float32(v * 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(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0));
	tmp = ((cosTheta_i * cosTheta_O) / t_0) - ((sinTheta_i * (cosTheta_i * (cosTheta_O * sinTheta_O))) / (v * t_0));
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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 64.2%

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

5. Taylor expanded in sinTheta_O around 0 64.2%

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

6. Final simplification 64.2%

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

Alternative 14: 65.2% accurate, 16.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{cosTheta_i \cdot cosTheta_O}{\frac{1}{v} \cdot 0.3333333333333333 + v \cdot 2} \end{array} \]

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 cosTheta_O) (+ (* (/ 1.0 v) 0.3333333333333333) (* v 2.0))))

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 * cosTheta_O) / (((1.0f / v) * 0.3333333333333333f) + (v * 2.0f));
}

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 * costheta_o) / (((1.0e0 / v) * 0.3333333333333333e0) + (v * 2.0e0))
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(Float32(cosTheta_i * cosTheta_O) / Float32(Float32(Float32(Float32(1.0) / v) * Float32(0.3333333333333333)) + Float32(v * Float32(2.0))))
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 * cosTheta_O) / (((single(1.0) / v) * single(0.3333333333333333)) + (v * single(2.0)));
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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 64.2%

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

5. Taylor expanded in sinTheta_O around 0 64.2%

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

6. Final simplification 64.2%

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

Alternative 15: 59.5% accurate, 31.4× speedup

Math

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

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
 (* 0.5 (/ cosTheta_i (/ 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 0.5f * (cosTheta_i / (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 = 0.5e0 * (costheta_i / (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(Float32(0.5) * Float32(cosTheta_i / 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 = single(0.5) * (cosTheta_i / (v / cosTheta_O));
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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 58.3%

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

   1. associate-/l* 58.2%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

6. Simplified 58.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}}} \]

7. Final simplification 58.2%

   \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O}} \]

Alternative 16: 59.5% accurate, 31.4× speedup

Math

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

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 cosTheta_O) v) 0.5))

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 * cosTheta_O) / v) * 0.5f;
}

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 * costheta_o) / v) * 0.5e0
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(cosTheta_i * cosTheta_O) / v) * Float32(0.5))
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 * cosTheta_O) / v) * single(0.5);
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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 58.3%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}} \]

5. Final simplification 58.3%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{v} \cdot 0.5 \]

Alternative 17: 60.0% accurate, 31.4× speedup

Math

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

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 (/ (/ v cosTheta_O) cosTheta_i)))

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 / ((v / cosTheta_O) / cosTheta_i);
}

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 / ((v / costheta_o) / costheta_i)
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(Float32(v / cosTheta_O) / cosTheta_i))
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) / ((v / cosTheta_O) / cosTheta_i);
end

Derivation

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/ 98.6%

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

   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* 98.8%

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

   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. Step-by-step derivation

   1. expm1-log1p-u 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)\right)} \]

   2. expm1-udef 53.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{cosTheta_i \cdot cosTheta_O}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right) \cdot e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)} - 1} \]

   3. times-frac 53.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}}\right)} - 1 \]

   4. *-commutative 53.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(v \cdot \left(v \cdot 2\right)\right)}} \cdot \frac{cosTheta_O}{e^{\frac{sinTheta_O}{v} \cdot sinTheta_i}}\right)} - 1 \]

   5. exp-prod 53.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \cdot \frac{cosTheta_O}{\color{blue}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}}\right)} - 1 \]

5. Applied egg-rr 53.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \cdot \frac{cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \cdot \frac{cosTheta_O}{{\left(e^{\frac{sinTheta_O}{v}}\right)}^{sinTheta_i}}\right)\right)} \]

   2. expm1-log1p 98.8%

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

7. Simplified 98.8%

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

8. Taylor expanded in v around inf 58.3%

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

   1. associate-*l/ 58.2%

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

   2. associate-*r* 58.2%

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

   3. associate-*r/ 58.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_i}{v}} \cdot cosTheta_O \]

   4. associate-/r/ 58.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_i}{\frac{v}{cosTheta_O}}} \]

   5. associate-/l* 58.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

10. Simplified 58.7%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}}} \]

11. Final simplification 58.7%

    \[\leadsto \frac{0.5}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \]

Reproduce

herbie shell --seed 2023229 
(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)))