HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 22.0s

Alternatives: 13

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: 98.8% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. associate-/l* 98.6%

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

   4. *-commutative 98.6%

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

   5. associate-*l/ 98.6%

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

5. Applied egg-rr 98.6%

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

   1. div-inv 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*r/ 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

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_i (- sinTheta_O)) 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_i * -sinTheta_O) / 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_i * -sintheta_o) / 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_i * Float32(-sinTheta_O)) / 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_i * -sinTheta_O) / v)) * ((single(1.0) / v) * (cosTheta_i * cosTheta_O))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.6%

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

   1. div-inv 98.7%

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

   2. *-commutative 98.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (*
   (/ cosTheta_O (exp (/ sinTheta_i (/ v sinTheta_O))))
   (/ (/ cosTheta_i v) v))
  (/ 0.5 (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_O / expf((sinTheta_i / (v / sinTheta_O)))) * ((cosTheta_i / v) / v)) * (0.5f / 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_o / exp((sintheta_i / (v / sintheta_o)))) * ((costheta_i / v) / v)) * (0.5e0 / 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(Float32(cosTheta_O / exp(Float32(sinTheta_i / Float32(v / sinTheta_O)))) * Float32(Float32(cosTheta_i / v) / v)) * Float32(Float32(0.5) / 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_O / exp((sinTheta_i / (v / sinTheta_O)))) * ((cosTheta_i / v) / v)) * (single(0.5) / sinh((single(1.0) / v)));
end

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. associate-/l* 98.6%

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

   4. *-commutative 98.6%

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

   5. associate-*l/ 98.6%

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

5. Applied egg-rr 98.6%

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

   1. div-inv 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*r/ 98.7%

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

7. Applied egg-rr 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. times-frac 98.6%

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

   3. un-div-inv 98.5%

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

   4. associate-/l* 98.5%

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

9. Applied egg-rr 98.5%

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

11. Simplified 98.5%

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

12. Final simplification 98.5%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

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 (/ (* v v) cosTheta_O))
   (exp (/ (* sinTheta_i sinTheta_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 ((cosTheta_i / ((v * v) / cosTheta_O)) / expf(((sinTheta_i * sinTheta_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 = ((costheta_i / ((v * v) / costheta_o)) / exp(((sintheta_i * sintheta_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(Float32(cosTheta_i / Float32(Float32(v * v) / cosTheta_O)) / exp(Float32(Float32(sinTheta_i * sinTheta_O) / 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 = ((cosTheta_i / ((v * v) / cosTheta_O)) / exp(((sinTheta_i * sinTheta_O) / v))) / (sinh((single(1.0) / v)) * single(2.0));
end

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   2. associate-*l/ 98.6%

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

   3. associate-/l/ 98.6%

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

   4. associate-*r/ 98.6%

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

   5. *-commutative 98.6%

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

   6. /-rgt-identity 98.6%

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

   7. associate-/r/ 98.6%

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

   8. exp-neg 98.6%

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

   9. remove-double-div 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-*l/ 98.6%

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

   12. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in cosTheta_i around 0 98.6%

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

   1. associate-/r* 98.6%

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

   2. associate-/l* 98.6%

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

   3. unpow2 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 5: 98.2% accurate, 1.8× speedup

Math

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

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 98.2%

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

   1. +-commutative 98.2%

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

   2. *-commutative 98.2%

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

   3. mul-1-neg 98.2%

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

   4. unsub-neg 98.2%

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

   5. associate-*l/ 98.2%

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

   6. *-commutative 98.2%

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

   7. associate-*r* 98.2%

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

   8. unpow2 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 6: 98.3% accurate, 1.9× speedup

Math

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

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 (sinh (/ 1.0 v))) (/ cosTheta_O (* v (/ v 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 / sinhf((1.0f / v))) * (cosTheta_O / (v * (v / 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 / sinh((1.0e0 / v))) * (costheta_o / (v * (v / 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(Float32(0.5) / sinh(Float32(Float32(1.0) / v))) * Float32(cosTheta_O / Float32(v * Float32(v / 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) / sinh((single(1.0) / v))) * (cosTheta_O / (v * (v / cosTheta_i)));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. associate-*r/ 97.9%

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

   3. associate-/l/ 98.1%

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

6. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.1%

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

   2. *-commutative 98.1%

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

   3. times-frac 98.0%

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

   4. associate-*r/ 98.0%

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

   5. associate-/r* 98.1%

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

   6. *-commutative 98.1%

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

   7. unpow2 98.1%

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

   8. associate-/l* 98.1%

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

   9. unpow2 98.1%

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

   10. associate-/l* 98.1%

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

   11. *-commutative 98.1%

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

   12. associate-/r* 98.1%

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

   13. metadata-eval 98.1%

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

8. Simplified 98.1%

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

   1. associate-/r/ 98.1%

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

10. Applied egg-rr 98.1%

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

11. Final simplification 98.1%

    \[\leadsto \frac{0.5}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta_O}{v \cdot \frac{v}{cosTheta_i}} \]

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

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 v) (/ (/ 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 (cosTheta_i / v) * ((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 = (costheta_i / v) * ((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(cosTheta_i / v) * Float32(Float32(cosTheta_O / 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 = (cosTheta_i / v) * ((cosTheta_O / v) / (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. associate-*r/ 97.9%

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

   3. associate-/l/ 98.1%

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

6. Applied egg-rr 98.1%

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

   1. *-lft-identity 98.1%

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

   2. times-frac 98.1%

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

   3. *-commutative 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 8: 98.3% accurate, 1.9× speedup

Math

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

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_O 0.5) (* (sinh (/ 1.0 v)) (* v (/ v 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 (cosTheta_O * 0.5f) / (sinhf((1.0f / v)) * (v * (v / 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 = (costheta_o * 0.5e0) / (sinh((1.0e0 / v)) * (v * (v / 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(cosTheta_O * Float32(0.5)) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v / 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 = (cosTheta_O * single(0.5)) / (sinh((single(1.0) / v)) * (v * (v / cosTheta_i)));
end

Derivation

1. Initial program 98.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. associate-*r/ 97.9%

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

   3. associate-/l/ 98.1%

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

6. Applied egg-rr 98.1%

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

   1. associate-*r/ 98.1%

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

   2. *-commutative 98.1%

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

   3. times-frac 98.0%

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

   4. associate-*r/ 98.0%

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

   5. associate-/r* 98.1%

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

   6. *-commutative 98.1%

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

   7. unpow2 98.1%

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

   8. associate-/l* 98.1%

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

   9. unpow2 98.1%

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

   10. associate-/l* 98.1%

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

   11. *-commutative 98.1%

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

   12. associate-/r* 98.1%

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

   13. metadata-eval 98.1%

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

8. Simplified 98.1%

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

   1. frac-times 98.2%

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

   2. div-inv 98.2%

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

   3. clear-num 98.2%

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

10. Applied egg-rr 98.2%

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

11. Final simplification 98.2%

    \[\leadsto \frac{cosTheta_O \cdot 0.5}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \frac{v}{cosTheta_i}\right)} \]

Alternative 9: 58.8% accurate, 24.4× speedup

Math

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

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 (/ 1.0 (/ v (* cosTheta_i 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 * (1.0f / (v / (cosTheta_i * 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 * (1.0e0 / (v / (costheta_i * 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(Float32(1.0) / Float32(v / Float32(cosTheta_i * 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) * (single(1.0) / (v / (cosTheta_i * cosTheta_O)));
end

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.9%

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

   1. associate-*r/ 57.8%

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

6. Simplified 57.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 57.9%

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

   2. clear-num 58.6%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}}} \]

8. Applied egg-rr 58.6%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}}} \]

9. Final simplification 58.6%

   \[\leadsto 0.5 \cdot \frac{1}{\frac{v}{cosTheta_i \cdot cosTheta_O}} \]

Alternative 10: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.9%

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

   1. associate-*l/ 57.8%

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

   2. *-commutative 57.8%

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

6. Simplified 57.8%

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

7. Final simplification 57.8%

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

Alternative 11: 58.4% 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 98.6%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.9%

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

   1. associate-*r/ 57.8%

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

6. Simplified 57.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O}{v}\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 57.9%

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

   2. associate-/l* 57.8%

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

8. Applied egg-rr 57.8%

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

9. Final simplification 57.8%

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

Alternative 12: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.5%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 57.9%

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

5. Final simplification 57.9%

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

Alternative 13: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.3%

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

4. Taylor expanded in v around inf 58.0%

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

5. Final simplification 58.0%

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

Reproduce

herbie shell --seed 2023275 
(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)))