HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.6%

Time: 14.7s

Alternatives: 15

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.4%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.5%

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

   4. associate-*r/ 98.5%

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

   5. *-commutative 98.5%

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

   6. /-rgt-identity 98.5%

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

   7. associate-/r/ 98.5%

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

   8. exp-neg 98.5%

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

   9. remove-double-div 98.5%

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

   10. *-commutative 98.5%

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

   11. associate-*l/ 98.5%

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

   12. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta_i around 0 98.5%

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

   1. associate-/r* 98.5%

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

   2. unpow2 98.5%

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

   3. times-frac 98.6%

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

   4. associate-/l* 98.6%

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

6. Simplified 98.6%

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

   1. associate-*l/ 98.6%

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

   2. associate-*r/ 98.5%

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

   3. associate-/l* 98.6%

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

8. Applied egg-rr 98.6%

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

9. Taylor expanded in cosTheta_O around 0 98.5%

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

    1. unpow2 98.5%

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

    2. times-frac 98.6%

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

    3. associate-*r/ 98.6%

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

    4. *-commutative 98.6%

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

    5. associate-/l* 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.4%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.5%

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

   4. associate-*r/ 98.5%

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

   5. *-commutative 98.5%

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

   6. /-rgt-identity 98.5%

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

   7. associate-/r/ 98.5%

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

   8. exp-neg 98.5%

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

   9. remove-double-div 98.5%

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

   10. *-commutative 98.5%

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

   11. associate-*l/ 98.5%

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

   12. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta_i around 0 98.5%

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

   1. associate-/r* 98.5%

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

   2. unpow2 98.5%

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

   3. times-frac 98.6%

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

   4. associate-/l* 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ cosTheta_i (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
  (/ (/ cosTheta_O 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 / (expf((1.0f / v)) - expf((-1.0f / v)))) * ((cosTheta_O / 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 / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * ((costheta_o / 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 / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(Float32(cosTheta_O / 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 / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * ((cosTheta_O / v) / v);
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.4%

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

   2. associate-*l/ 98.5%

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

   3. associate-/l/ 98.5%

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

   4. associate-*r/ 98.5%

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

   5. *-commutative 98.5%

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

   6. /-rgt-identity 98.5%

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

   7. associate-/r/ 98.5%

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

   8. exp-neg 98.5%

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

   9. remove-double-div 98.5%

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

   10. *-commutative 98.5%

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

   11. associate-*l/ 98.5%

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

   12. exp-prod 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in cosTheta_i around 0 98.5%

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

   1. associate-/r* 98.5%

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

   2. unpow2 98.5%

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

   3. times-frac 98.6%

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

   4. associate-/l* 98.6%

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

6. Simplified 98.6%

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

   1. associate-*l/ 98.6%

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

   2. associate-*r/ 98.5%

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

   3. associate-/l* 98.6%

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

8. Applied egg-rr 98.6%

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

9. Taylor expanded in sinTheta_O around 0 98.0%

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

    1. times-frac 98.0%

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

    2. rec-exp 98.0%

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

    3. distribute-neg-frac 98.0%

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

    4. metadata-eval 98.0%

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

    5. *-commutative 98.0%

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

    6. unpow2 98.0%

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

    7. associate-/r* 98.1%

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

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

Alternative 4: 98.2% accurate, 1.8× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (* cosTheta_O (/ cosTheta_i v))))
   (/
    (/
     (- t_0 (* (/ sinTheta_O (/ v sinTheta_i)) t_0))
     (* (sinh (/ 1.0 v)) 2.0))
    v)))

C

assert(cosTheta_i < cosTheta_O);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cosTheta_O * (cosTheta_i / v);
	return ((t_0 - ((sinTheta_O / (v / sinTheta_i)) * t_0)) / (sinhf((1.0f / v)) * 2.0f)) / v;
}

Fortran

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = costheta_o * (costheta_i / v)
    code = ((t_0 - ((sintheta_o / (v / sintheta_i)) * t_0)) / (sinh((1.0e0 / v)) * 2.0e0)) / v
end function

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(cosTheta_O * Float32(cosTheta_i / v))
	return Float32(Float32(Float32(t_0 - Float32(Float32(sinTheta_O / Float32(v / sinTheta_i)) * t_0)) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))) / v)
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = cosTheta_O * (cosTheta_i / v);
	tmp = ((t_0 - ((sinTheta_O / (v / sinTheta_i)) * t_0)) / (sinh((single(1.0) / v)) * single(2.0))) / v;
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 98.0%

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

   1. +-commutative 98.0%

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

   2. mul-1-neg 98.0%

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

   3. unsub-neg 98.0%

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

   4. *-commutative 98.0%

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

   5. associate-*l/ 98.1%

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

   6. *-commutative 98.1%

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

   7. associate-*r* 98.1%

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

   8. unpow2 98.1%

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

   9. times-frac 98.1%

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

   10. *-commutative 98.1%

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

   11. associate-*l/ 98.1%

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

   12. *-commutative 98.1%

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

   13. associate-/l* 98.1%

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

6. Simplified 98.1%

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

7. Final simplification 98.1%

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

Alternative 5: 67.2% accurate, 1.9× speedup

Math

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

FPCore

NOTE: cosTheta_i and cosTheta_O should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ cosTheta_O v) (/ cosTheta_i (+ (exp (/ 1.0 v)) (+ (/ 1.0 v) -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 / (expf((1.0f / v)) + ((1.0f / v) + -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 / (exp((1.0e0 / v)) + ((1.0e0 / v) + (-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 / Float32(exp(Float32(Float32(1.0) / v)) + Float32(Float32(Float32(1.0) / v) + 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 / (exp((single(1.0) / v)) + ((single(1.0) / v) + single(-1.0))))) / v;
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.2%

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

4. Taylor expanded in sinTheta_i around 0 97.6%

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

   1. times-frac 97.6%

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

   2. rec-exp 97.6%

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

   3. distribute-neg-frac 97.6%

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

   4. metadata-eval 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in v around inf 66.6%

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

8. Final simplification 66.6%

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

Alternative 6: 67.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in sinTheta_i around 0 98.0%

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

   1. associate-/r* 98.0%

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

   2. unpow2 98.0%

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

   3. rec-exp 98.0%

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

   4. distribute-neg-frac 98.0%

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

   5. metadata-eval 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in v around inf 66.7%

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

8. Final simplification 66.7%

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

Alternative 7: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O])\\ \\ \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}}}{v} \end{array} \]

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_O v)
   (/ cosTheta_i (+ (/ 2.0 v) (/ 0.3333333333333333 (pow v 3.0)))))
  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 / ((2.0f / v) + (0.3333333333333333f / powf(v, 3.0f))))) / 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 / ((2.0e0 / v) + (0.3333333333333333e0 / (v ** 3.0e0))))) / 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 / Float32(Float32(Float32(2.0) / v) + Float32(Float32(0.3333333333333333) / (v ^ Float32(3.0)))))) / 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 / ((single(2.0) / v) + (single(0.3333333333333333) / (v ^ single(3.0)))))) / v;
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.2%

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

4. Taylor expanded in sinTheta_i around 0 97.6%

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

   1. times-frac 97.6%

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

   2. rec-exp 97.6%

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

   3. distribute-neg-frac 97.6%

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

   4. metadata-eval 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in v around inf 62.7%

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

   1. associate-*r/ 62.7%

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

   2. metadata-eval 62.7%

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

   3. associate-*r/ 62.7%

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

   4. metadata-eval 62.7%

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

9. Simplified 62.7%

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

10. Final simplification 62.7%

    \[\leadsto \frac{\frac{cosTheta_O}{v} \cdot \frac{cosTheta_i}{\frac{2}{v} + \frac{0.3333333333333333}{{v}^{3}}}}{v} \]

Alternative 8: 64.2% accurate, 6.7× speedup

Math

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

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_O (/ t_0 (/ cosTheta_i v)))
    (*
     (* (/ cosTheta_O v) (/ cosTheta_i v))
     (/ (* sinTheta_O sinTheta_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_O / (t_0 / (cosTheta_i / v))) - (((cosTheta_O / v) * (cosTheta_i / v)) * ((sinTheta_O * sinTheta_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_o / (t_0 / (costheta_i / v))) - (((costheta_o / v) * (costheta_i / v)) * ((sintheta_o * sintheta_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(cosTheta_O / Float32(t_0 / Float32(cosTheta_i / v))) - Float32(Float32(Float32(cosTheta_O / v) * Float32(cosTheta_i / v)) * Float32(Float32(sinTheta_O * sinTheta_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_O / (t_0 / (cosTheta_i / v))) - (((cosTheta_O / v) * (cosTheta_i / v)) * ((sinTheta_O * sinTheta_i) / t_0));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in v around inf 62.6%

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

   1. associate-*r/ 62.6%

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

   2. metadata-eval 62.6%

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

   3. unpow2 62.6%

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

4. Simplified 62.6%

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

5. Taylor expanded in sinTheta_i around 0 62.6%

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

   1. +-commutative 62.6%

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

   2. mul-1-neg 62.6%

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

   3. unsub-neg 62.6%

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

7. Simplified 62.6%

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

8. Final simplification 62.6%

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

Alternative 9: 59.1% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(v / Float32(Float32(cosTheta_i * Float32(cosTheta_O / v)) / Float32(Float32(2.0) / v))))
end

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.2%

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

4. Taylor expanded in sinTheta_i around 0 97.6%

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

   1. times-frac 97.6%

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

   2. rec-exp 97.6%

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

   3. distribute-neg-frac 97.6%

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

   4. metadata-eval 97.6%

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

6. Simplified 97.6%

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

   1. clear-num 92.0%

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

   2. inv-pow 92.0%

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

8. Applied egg-rr 92.0%

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

   1. unpow-1 92.0%

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

   2. associate-*r/ 92.0%

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

   3. *-commutative 92.0%

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

10. Simplified 92.0%

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

11. Taylor expanded in v around inf 57.5%

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

12. Final simplification 57.5%

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

Alternative 10: 64.2% accurate, 16.9× speedup

Math

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

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_O (/ (+ 2.0 (/ 0.3333333333333333 (* v v))) (/ cosTheta_i v))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in v around inf 62.6%

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

   1. associate-*r/ 62.6%

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

   2. metadata-eval 62.6%

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

   3. unpow2 62.6%

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

4. Simplified 62.6%

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

5. Taylor expanded in sinTheta_i around 0 62.6%

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

   1. associate-/l* 62.6%

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

   2. *-commutative 62.6%

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

   3. associate-/l* 62.6%

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

   4. *-rgt-identity 62.6%

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

   5. associate-*r/ 62.6%

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

   6. associate-*r/ 62.6%

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

   7. metadata-eval 62.6%

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

   8. unpow2 62.6%

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

   9. associate-*r/ 62.6%

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

   10. *-rgt-identity 62.6%

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

7. Simplified 62.6%

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

8. Final simplification 62.6%

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

Alternative 11: 59.0% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

cosTheta_i, cosTheta_O = sort([cosTheta_i, cosTheta_O])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(Float32(1.0) / Float32(v / Float32(cosTheta_i * cosTheta_O))))
end

MATLAB

cosTheta_i, cosTheta_O = num2cell(sort([cosTheta_i, cosTheta_O])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (single(1.0) / (v / (cosTheta_i * cosTheta_O)));
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 56.7%

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

   1. *-commutative 56.7%

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

   2. associate-*l/ 56.7%

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

   3. *-commutative 56.7%

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

6. Simplified 56.7%

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

   1. associate-*r/ 56.7%

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

   2. clear-num 57.4%

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

8. Applied egg-rr 57.4%

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

9. Final simplification 57.4%

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

Alternative 12: 58.5% accurate, 31.4× speedup

Math

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

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_O (/ cosTheta_i 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_O * (cosTheta_i / 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_o * (costheta_i / 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_O * Float32(cosTheta_i / 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_O * (cosTheta_i / v)) * single(0.5);
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 56.7%

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

   1. *-commutative 56.7%

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

   2. associate-*l/ 56.7%

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

   3. *-commutative 56.7%

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

6. Simplified 56.7%

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

7. Final simplification 56.7%

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

Alternative 13: 58.5% 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 98.5%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 56.7%

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

   1. associate-*l/ 56.7%

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

   2. *-commutative 56.7%

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

6. Simplified 56.7%

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

7. Final simplification 56.7%

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

Alternative 14: 58.5% accurate, 31.4× speedup

Math

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

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 98.5%

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

   1. times-frac 98.4%

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

   2. *-commutative 98.4%

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

   3. associate-/l* 98.4%

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

   4. associate-/l/ 98.5%

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

   5. distribute-neg-frac 98.5%

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

   6. distribute-rgt-neg-out 98.5%

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

   7. associate-*l/ 98.5%

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

3. Simplified 98.5%

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

4. Taylor expanded in v around inf 56.7%

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

5. Final simplification 56.7%

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

Alternative 15: 58.5% 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 98.5%

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

3. Simplified 98.2%

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

4. Taylor expanded in v around inf 56.7%

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

5. Final simplification 56.7%

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

Reproduce

herbie shell --seed 2023293 
(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)))