HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 14.3s

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

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

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
 (/
  (*
   (/ (exp (/ (* sinTheta_O (- sinTheta_i)) v)) v)
   (* cosTheta_i (/ cosTheta_O v)))
  (* 2.0 (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 ((expf(((sinTheta_O * -sinTheta_i) / v)) / v) * (cosTheta_i * (cosTheta_O / v))) / (2.0f * 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 = ((exp(((sintheta_o * -sintheta_i) / v)) / v) * (costheta_i * (costheta_o / v))) / (2.0e0 * 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(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) / v) * Float32(cosTheta_i * Float32(cosTheta_O / v))) / Float32(Float32(2.0) * 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 = ((exp(((sinTheta_O * -sinTheta_i) / v)) / v) * (cosTheta_i * (cosTheta_O / v))) / (single(2.0) * 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. distribute-neg-frac 98.7%

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

   2. *-commutative 98.7%

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

   3. distribute-rgt-neg-in 98.7%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

5. Applied egg-rr 98.5%

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

   1. associate-*r/ 98.5%

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

   2. associate-*r/ 98.9%

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

   3. associate-*r/ 98.9%

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

   4. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * (costheta_o / v)) * (((0.5e0 / sinh((1.0e0 / v))) / v) / exp((sintheta_i / (v / sintheta_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(cosTheta_i * Float32(cosTheta_O / v)) * Float32(Float32(Float32(Float32(0.5) / sinh(Float32(Float32(1.0) / v))) / v) / exp(Float32(sinTheta_i / Float32(v / sinTheta_O)))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * (cosTheta_O / v)) * (((single(0.5) / sinh((single(1.0) / v))) / v) / exp((sinTheta_i / (v / sinTheta_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. *-commutative 98.7%

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

   2. associate-*r/ 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-*r* 98.6%

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

   8. associate-/l/ 98.6%

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

   9. exp-neg 98.6%

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

   10. associate-/l/ 98.6%

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

   11. associate-/r* 98.6%

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

   12. metadata-eval 98.6%

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

   13. associate-*l/ 98.6%

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

   14. *-commutative 98.6%

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

   15. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in sinTheta_O around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. rec-exp 98.7%

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

   3. distribute-neg-frac 98.7%

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

   4. metadata-eval 98.7%

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

6. Simplified 98.7%

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

   1. inv-pow 98.7%

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

   2. associate-*r* 98.7%

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

   3. div-inv 98.7%

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

   4. neg-mul-1 98.7%

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

   5. sinh-undef 98.6%

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

   6. unpow-prod-down 98.9%

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

   7. associate-*r/ 98.9%

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

   8. inv-pow 98.9%

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

8. Applied egg-rr 98.9%

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

   1. associate-*r/ 99.0%

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

   2. *-rgt-identity 99.0%

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

   3. unpow-1 99.0%

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

   4. associate-/r* 98.6%

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

   5. associate-*r* 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-/r* 98.6%

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

   8. *-commutative 98.6%

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

   9. associate-/r* 98.7%

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

   10. associate-/r* 98.7%

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

   11. metadata-eval 98.7%

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

   12. associate-/l* 98.7%

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

10. Simplified 98.7%

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

11. Final simplification 98.7%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * costheta_o) / ((v * (2.0e0 * (v * sinh((1.0e0 / v))))) * exp((sintheta_i * (sintheta_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(cosTheta_i * cosTheta_O) / Float32(Float32(v * Float32(Float32(2.0) * Float32(v * sinh(Float32(Float32(1.0) / v))))) * exp(Float32(sinTheta_i * Float32(sinTheta_O / v)))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) / ((v * (single(2.0) * (v * sinh((single(1.0) / v))))) * exp((sinTheta_i * (sinTheta_O / 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 98.8%

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

   3. exp-neg 98.8%

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

   4. associate-*l/ 98.8%

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

   5. *-lft-identity 98.8%

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

   6. associate-/l/ 98.8%

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

   7. associate-*l* 98.8%

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

   8. associate-*l* 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

   1. add-exp-log 98.5%

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

5. Applied egg-rr 98.5%

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

   1. expm1-log1p-u 98.4%

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

   2. expm1-udef 98.4%

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

   3. add-exp-log 98.7%

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

   4. *-commutative 98.7%

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

7. Applied egg-rr 98.7%

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

   1. expm1-def 98.7%

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

   2. expm1-log1p 98.7%

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

   3. *-commutative 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l* 98.8%

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

   6. associate-*l* 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

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 (/ cosTheta_O v))
  (/ (/ 1.0 v) (- (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 * (cosTheta_O / v)) * ((1.0f / v) / (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 * (costheta_o / v)) * ((1.0e0 / v) / (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(cosTheta_O / v)) * Float32(Float32(Float32(1.0) / v) / 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 * (cosTheta_O / v)) * ((single(1.0) / v) / (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. *-commutative 98.7%

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

   2. associate-*r/ 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l/ 98.6%

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

   5. *-commutative 98.6%

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

   6. *-commutative 98.6%

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

   7. associate-*r* 98.6%

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

   8. associate-/l/ 98.6%

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

   9. exp-neg 98.6%

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

   10. associate-/l/ 98.6%

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

   11. associate-/r* 98.6%

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

   12. metadata-eval 98.6%

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

   13. associate-*l/ 98.6%

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

   14. *-commutative 98.6%

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

   15. exp-prod 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in sinTheta_O around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. rec-exp 98.7%

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

   3. distribute-neg-frac 98.7%

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

   4. metadata-eval 98.7%

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

6. Simplified 98.7%

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

7. Taylor expanded in sinTheta_i around 0 98.1%

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

   1. associate-/l/ 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

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_i (pow (/ cosTheta_O (* v v)) -1.0)) (* 2.0 (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 / powf((cosTheta_O / (v * v)), -1.0f)) / (2.0f * 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 / (v * v)) ** (-1.0e0))) / (2.0e0 * 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 / (Float32(cosTheta_O / Float32(v * v)) ^ Float32(-1.0))) / Float32(Float32(2.0) * 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 / (v * v)) ^ single(-1.0))) / (single(2.0) * 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. distribute-neg-frac 98.7%

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

   2. *-commutative 98.7%

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

   3. distribute-rgt-neg-in 98.7%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

5. Applied egg-rr 98.5%

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

6. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-/l* 98.1%

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

   3. unpow2 98.1%

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

   4. rec-exp 98.1%

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

8. Simplified 98.1%

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

   1. clear-num 98.2%

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

   2. inv-pow 98.2%

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

10. Applied egg-rr 98.2%

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

    1. sinh-undef 98.2%

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

    2. *-commutative 98.2%

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

12. Applied egg-rr 98.2%

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

13. Final simplification 98.2%

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

Alternative 6: 98.3% accurate, 1.9× speedup

Math

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

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 v) (/ cosTheta_i v)) (* 2.0 (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 / v) * (cosTheta_i / v)) / (2.0f * 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 / v) * (costheta_i / v)) / (2.0e0 * 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 / v) * Float32(cosTheta_i / v)) / Float32(Float32(2.0) * 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 / v) * (cosTheta_i / v)) / (single(2.0) * 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. distribute-neg-frac 98.7%

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

   2. *-commutative 98.7%

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

   3. distribute-rgt-neg-in 98.7%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

5. Applied egg-rr 98.5%

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

   1. associate-*r/ 98.5%

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

   2. associate-*r/ 98.9%

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

   3. associate-*r/ 98.9%

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

   4. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in sinTheta_O around 0 98.0%

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

   1. *-commutative 98.0%

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

   2. unpow2 98.0%

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

   3. times-frac 98.0%

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

10. Simplified 98.0%

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

11. Final simplification 98.0%

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

Alternative 7: 98.3% accurate, 1.9× speedup

Math

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

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_i (* v (/ v cosTheta_O))) (* 2.0 (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 / (v * (v / cosTheta_O))) / (2.0f * 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 / (v * (v / costheta_o))) / (2.0e0 * 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 / Float32(v * Float32(v / cosTheta_O))) / Float32(Float32(2.0) * 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 / (v * (v / cosTheta_O))) / (single(2.0) * 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. distribute-neg-frac 98.7%

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

   2. *-commutative 98.7%

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

   3. distribute-rgt-neg-in 98.7%

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

   4. associate-*l/ 98.7%

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

   5. *-commutative 98.7%

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

3. Simplified 98.7%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

5. Applied egg-rr 98.5%

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

6. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-/l* 98.1%

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

   3. unpow2 98.1%

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

   4. rec-exp 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in v around 0 98.1%

   \[\leadsto \frac{\frac{cosTheta_i}{\color{blue}{\frac{{v}^{2}}{cosTheta_O}}}}{e^{\frac{1}{v}} - e^{-\frac{1}{v}}} \]
10. Step-by-step derivation

    1. unpow2 98.1%

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

    2. associate-*l/ 98.1%

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

    3. *-commutative 98.1%

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

11. Simplified 98.1%

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

    1. sinh-undef 98.2%

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

    2. *-commutative 98.2%

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

13. Applied egg-rr 98.2%

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

14. Final simplification 98.2%

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

Alternative 8: 64.5% accurate, 6.7× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 2.0 (/ 0.3333333333333333 (* v v)))))
   (-
    (* (/ cosTheta_i v) (/ cosTheta_O t_0))
    (*
     (/ sinTheta_i (* v v))
     (/ (* cosTheta_O (* sinTheta_O cosTheta_i)) 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 = 2.0f + (0.3333333333333333f / (v * v));
	return ((cosTheta_i / v) * (cosTheta_O / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_O * (sinTheta_O * cosTheta_i)) / 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 = 2.0e0 + (0.3333333333333333e0 / (v * v))
    code = ((costheta_i / v) * (costheta_o / t_0)) - ((sintheta_i / (v * v)) * ((costheta_o * (sintheta_o * costheta_i)) / 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(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))
	return Float32(Float32(Float32(cosTheta_i / v) * Float32(cosTheta_O / t_0)) - Float32(Float32(sinTheta_i / Float32(v * v)) * Float32(Float32(cosTheta_O * Float32(sinTheta_O * cosTheta_i)) / 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(2.0) + (single(0.3333333333333333) / (v * v));
	tmp = ((cosTheta_i / v) * (cosTheta_O / t_0)) - ((sinTheta_i / (v * v)) * ((cosTheta_O * (sinTheta_O * cosTheta_i)) / 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. Taylor expanded in v around inf 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.8%

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

   2. metadata-eval 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

   3. unpow2 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

4. Simplified 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

5. Taylor expanded in sinTheta_i around 0 64.8%

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

   1. times-frac 64.8%

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

   2. *-rgt-identity 64.8%

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

   3. *-rgt-identity 64.8%

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

   4. associate-*r/ 64.8%

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

   5. metadata-eval 64.8%

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

   6. unpow2 64.8%

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

   7. mul-1-neg 64.8%

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

   8. times-frac 64.8%

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

   9. unpow2 64.8%

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

   10. associate-*r* 64.8%

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

   11. associate-*r/ 64.8%

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

7. Simplified 64.8%

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

8. Final simplification 64.8%

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

Alternative 9: 64.5% accurate, 6.7× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 2.0 (/ 0.3333333333333333 (* v v)))))
   (-
    (/ (* cosTheta_i (/ cosTheta_O v)) t_0)
    (*
     (/ sinTheta_i (* v v))
     (/ (* cosTheta_O (* sinTheta_O cosTheta_i)) 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 = 2.0f + (0.3333333333333333f / (v * v));
	return ((cosTheta_i * (cosTheta_O / v)) / t_0) - ((sinTheta_i / (v * v)) * ((cosTheta_O * (sinTheta_O * cosTheta_i)) / 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 = 2.0e0 + (0.3333333333333333e0 / (v * v))
    code = ((costheta_i * (costheta_o / v)) / t_0) - ((sintheta_i / (v * v)) * ((costheta_o * (sintheta_o * costheta_i)) / 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(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v)))
	return Float32(Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) / t_0) - Float32(Float32(sinTheta_i / Float32(v * v)) * Float32(Float32(cosTheta_O * Float32(sinTheta_O * cosTheta_i)) / 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(2.0) + (single(0.3333333333333333) / (v * v));
	tmp = ((cosTheta_i * (cosTheta_O / v)) / t_0) - ((sinTheta_i / (v * v)) * ((cosTheta_O * (sinTheta_O * cosTheta_i)) / 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. Taylor expanded in v around inf 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.8%

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

   2. metadata-eval 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

   3. unpow2 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

4. Simplified 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

5. Taylor expanded in sinTheta_i around 0 64.8%

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

   1. times-frac 64.8%

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

   2. *-rgt-identity 64.8%

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

   3. *-rgt-identity 64.8%

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

   4. associate-*r/ 64.8%

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

   5. metadata-eval 64.8%

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

   6. unpow2 64.8%

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

   7. mul-1-neg 64.8%

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

   8. times-frac 64.8%

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

   9. unpow2 64.8%

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

   10. associate-*r* 64.8%

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

   11. associate-*r/ 64.8%

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

7. Simplified 64.8%

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

8. Taylor expanded in cosTheta_i around 0 64.8%

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

   1. associate-/r* 64.8%

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

   2. associate-*r/ 64.8%

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

   3. unpow2 64.8%

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

   4. associate-*r/ 64.8%

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

   5. metadata-eval 64.8%

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

10. Simplified 64.8%

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

11. Final simplification 64.8%

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

Alternative 10: 64.5% accurate, 16.9× speedup

Math

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

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) (/ cosTheta_O (+ 2.0 (/ 0.3333333333333333 (* v 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) * (cosTheta_O / (2.0f + (0.3333333333333333f / (v * 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) * (costheta_o / (2.0e0 + (0.3333333333333333e0 / (v * 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 / v) * Float32(cosTheta_O / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * 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) * (cosTheta_O / (single(2.0) + (single(0.3333333333333333) / (v * 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. Taylor expanded in v around inf 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.8%

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

   2. metadata-eval 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

   3. unpow2 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

4. Simplified 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

5. Taylor expanded in sinTheta_i around 0 64.7%

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

   1. times-frac 64.7%

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

   2. *-rgt-identity 64.7%

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

   3. *-rgt-identity 64.7%

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

   4. associate-*r/ 64.7%

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

   5. metadata-eval 64.7%

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

   6. unpow2 64.7%

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

7. Simplified 64.7%

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

8. Final simplification 64.7%

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

Alternative 11: 58.9% accurate, 31.4× speedup

Math

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

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(cosTheta_i * Float32(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. Taylor expanded in v around inf 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}} \]
3. Step-by-step derivation

   1. associate-*r/ 64.8%

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

   2. metadata-eval 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}} \]

   3. unpow2 64.8%

      \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{2 + \frac{0.3333333333333333}{\color{blue}{v \cdot v}}} \]

4. Simplified 64.8%

   \[\leadsto \frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\color{blue}{2 + \frac{0.3333333333333333}{v \cdot v}}} \]

5. Taylor expanded in v around inf 58.2%

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

   1. associate-/l* 58.2%

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

7. Simplified 58.2%

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

   1. div-inv 58.2%

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

   2. clear-num 58.2%

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

   3. *-commutative 58.2%

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

9. Applied egg-rr 58.2%

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

10. Final simplification 58.2%

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

Alternative 12: 58.9% 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 98.7%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.8%

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

   3. exp-neg 98.8%

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

   4. associate-*l/ 98.8%

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

   5. *-lft-identity 98.8%

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

   6. associate-/l/ 98.8%

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

   7. associate-*l* 98.8%

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

   8. associate-*l* 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 58.2%

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

5. Final simplification 58.2%

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

Alternative 13: 58.9% accurate, 31.4× speedup

Math

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

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) 0.5) 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) * 0.5f) / 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) * 0.5e0) / 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_i * cosTheta_O) * Float32(0.5)) / 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) * single(0.5)) / 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 98.8%

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

   3. exp-neg 98.8%

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

   4. associate-*l/ 98.8%

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

   5. *-lft-identity 98.8%

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

   6. associate-/l/ 98.8%

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

   7. associate-*l* 98.8%

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

   8. associate-*l* 98.7%

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

   9. *-commutative 98.7%

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

   10. *-commutative 98.7%

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

   11. associate-*l/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in v around inf 58.2%

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

   1. associate-*r/ 58.2%

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

6. Applied egg-rr 58.2%

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

7. Final simplification 58.2%

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

Reproduce

herbie shell --seed 2023200 
(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)))