HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 17.9s

Alternatives: 12

Speedup: 1.9×

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: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i \cdot \left(\left(\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

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

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v 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 * ((((1.0e0 - (sintheta_o * (sintheta_i / v))) / v) * costheta_o) * ((0.5e0 / v) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) \cdot \color{blue}{\frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}\right) \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. exp-neg N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   8. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{\color{blue}{1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   9. associate-*l* N/A

      \[\leadsto cosTheta\_i \cdot \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{\frac{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{cosTheta\_O}}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{v}\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \left(\frac{v}{sinTheta\_O}\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   9. /-lowering-/.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto cosTheta\_i \cdot \left(\color{blue}{\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right)} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

7. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}\right)}, cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \left(\frac{sinTheta\_i}{v}\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto cosTheta\_i \cdot \left(\left(\color{blue}{\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v}} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]
10. Add Preprocessing

Alternative 2: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) \cdot \color{blue}{\frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}\right) \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. exp-neg N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   8. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{\color{blue}{1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   9. associate-*l* N/A

      \[\leadsto cosTheta\_i \cdot \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{\frac{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{cosTheta\_O}}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{v}\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \left(\frac{v}{sinTheta\_O}\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   9. /-lowering-/.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto cosTheta\_i \cdot \left(\color{blue}{\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right)} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

7. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(\frac{1}{v}\right)}, cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 98.8%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

9. Simplified 98.8%

   \[\leadsto cosTheta\_i \cdot \left(\left(\color{blue}{\frac{1}{v}} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

10. Final simplification 98.8%

    \[\leadsto cosTheta\_i \cdot \left(\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \left(cosTheta\_O \cdot \frac{1}{v}\right)\right) \]
11. Add Preprocessing

Alternative 3: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) \cdot \color{blue}{\frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}\right) \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. exp-neg N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   8. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{\color{blue}{1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   9. associate-*l* N/A

      \[\leadsto cosTheta\_i \cdot \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\color{blue}{\left(\frac{cosTheta\_O}{v}\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 98.6%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, v\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

7. Simplified 98.6%

   \[\leadsto cosTheta\_i \cdot \left(\color{blue}{\frac{cosTheta\_O}{v}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

8. Final simplification 98.6%

   \[\leadsto cosTheta\_i \cdot \left(\frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)} \cdot \frac{cosTheta\_O}{v}\right) \]
9. Add Preprocessing

Alternative 4: 70.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i \cdot \left(\left(sinTheta\_i \cdot \left(\frac{cosTheta\_O}{sinTheta\_i \cdot v} - cosTheta\_O \cdot \frac{sinTheta\_O}{v \cdot v}\right)\right) \cdot \frac{\frac{0.5}{v}}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}\right) \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i
  (*
   (*
    sinTheta_i
    (- (/ cosTheta_O (* sinTheta_i v)) (* cosTheta_O (/ sinTheta_O (* v v)))))
   (/
    (/ 0.5 v)
    (/
     (-
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      -1.0)
     v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * ((sinTheta_i * ((cosTheta_O / (sinTheta_i * v)) - (cosTheta_O * (sinTheta_O / (v * v))))) * ((0.5f / v) / ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f) / v)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v 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 * ((sintheta_i * ((costheta_o / (sintheta_i * v)) - (costheta_o * (sintheta_o / (v * v))))) * ((0.5e0 / v) / ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(Float32(sinTheta_i * Float32(Float32(cosTheta_O / Float32(sinTheta_i * v)) - Float32(cosTheta_O * Float32(sinTheta_O / Float32(v * v))))) * Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) - Float32(-1.0)) / v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * ((sinTheta_i * ((cosTheta_O / (sinTheta_i * v)) - (cosTheta_O * (sinTheta_O / (v * v))))) * ((single(0.5) / v) / ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - single(-1.0)) / v)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) \cdot \color{blue}{\frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}\right) \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. exp-neg N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   8. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{\color{blue}{1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   9. associate-*l* N/A

      \[\leadsto cosTheta\_i \cdot \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\color{blue}{\left(\frac{cosTheta\_O + -1 \cdot \frac{cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}{v}\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O + -1 \cdot \frac{cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right), v\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(cosTheta\_O, \left(-1 \cdot \frac{cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(cosTheta\_O, \left(\frac{-1 \cdot \left(cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\left(-1 \cdot \left(cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(-1, \left(cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(-1, \mathsf{*.f32}\left(cosTheta\_O, \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), v\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 98.9%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(\mathsf{*.f32}\left(-1, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right), v\right)\right), v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

7. Simplified 98.9%

   \[\leadsto cosTheta\_i \cdot \left(\color{blue}{\frac{cosTheta\_O + \frac{-1 \cdot \left(cosTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}}{v}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

8. Taylor expanded in sinTheta_i around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\color{blue}{\left(sinTheta\_i \cdot \left(-1 \cdot \frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}} + \frac{cosTheta\_O}{sinTheta\_i \cdot v}\right)\right)}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \left(-1 \cdot \frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}} + \frac{cosTheta\_O}{sinTheta\_i \cdot v}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \left(\left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}}\right)\right) + \frac{cosTheta\_O}{sinTheta\_i \cdot v}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \left(\frac{cosTheta\_O}{sinTheta\_i \cdot v} + \left(\mathsf{neg}\left(\frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \left(\frac{cosTheta\_O}{sinTheta\_i \cdot v} - \frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}}\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\left(\frac{cosTheta\_O}{sinTheta\_i \cdot v}\right), \left(\frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(sinTheta\_i \cdot v\right)\right), \left(\frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \left(v \cdot sinTheta\_i\right)\right), \left(\frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \left(\frac{cosTheta\_O \cdot sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \left(cosTheta\_O \cdot \frac{sinTheta\_O}{{v}^{2}}\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{sinTheta\_O}{{v}^{2}}\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(sinTheta\_O, \left({v}^{2}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(sinTheta\_O, \left(v \cdot v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 98.8%

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

10. Simplified 98.8%

    \[\leadsto cosTheta\_i \cdot \left(\color{blue}{\left(sinTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot sinTheta\_i} - cosTheta\_O \cdot \frac{sinTheta\_O}{v \cdot v}\right)\right)} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

11. Taylor expanded in v around -inf

    \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}\right)\right)\right) \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)\right)\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{\mathsf{neg}\left(v\right)}}\right)\right)\right)\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(sinTheta\_i, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(v, sinTheta\_i\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(sinTheta\_O, \mathsf{*.f32}\left(v, v\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), \color{blue}{\left(\mathsf{neg}\left(v\right)\right)}\right)\right)\right)\right) \]

13. Simplified 69.9%

    \[\leadsto cosTheta\_i \cdot \left(\left(sinTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot sinTheta\_i} - cosTheta\_O \cdot \frac{sinTheta\_O}{v \cdot v}\right)\right) \cdot \frac{\frac{0.5}{v}}{\color{blue}{\frac{-1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v}}{-v}}}\right) \]

14. Final simplification 69.9%

    \[\leadsto cosTheta\_i \cdot \left(\left(sinTheta\_i \cdot \left(\frac{cosTheta\_O}{sinTheta\_i \cdot v} - cosTheta\_O \cdot \frac{sinTheta\_O}{v \cdot v}\right)\right) \cdot \frac{\frac{0.5}{v}}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}\right) \]
15. Add Preprocessing

Alternative 5: 70.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i \cdot \left(\left(\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}\right) \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i
  (*
   (* (/ (- 1.0 (* sinTheta_O (/ sinTheta_i v))) v) cosTheta_O)
   (/
    (/ 0.5 v)
    (/
     (-
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      -1.0)
     v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * ((((1.0f - (sinTheta_O * (sinTheta_i / v))) / v) * cosTheta_O) * ((0.5f / v) / ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f) / v)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v 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 * ((((1.0e0 - (sintheta_o * (sintheta_i / v))) / v) * costheta_o) * ((0.5e0 / v) / ((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(Float32(Float32(Float32(Float32(1.0) - Float32(sinTheta_O * Float32(sinTheta_i / v))) / v) * cosTheta_O) * Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) - Float32(-1.0)) / v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * ((((single(1.0) - (sinTheta_O * (sinTheta_i / v))) / v) * cosTheta_O) * ((single(0.5) / v) / ((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - single(-1.0)) / v)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) \cdot \color{blue}{\frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}\right) \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. exp-neg N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   8. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{\color{blue}{1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   9. associate-*l* N/A

      \[\leadsto cosTheta\_i \cdot \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{\frac{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{cosTheta\_O}}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{v}\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \left(\frac{v}{sinTheta\_O}\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   9. /-lowering-/.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto cosTheta\_i \cdot \left(\color{blue}{\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right)} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

7. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}\right)}, cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \left(\frac{sinTheta\_i}{v}\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto cosTheta\_i \cdot \left(\left(\color{blue}{\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v}} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

10. Taylor expanded in v around -inf

    \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)\right)\right)\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{\mathsf{neg}\left(v\right)}}\right)\right)\right)\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), \color{blue}{\left(\mathsf{neg}\left(v\right)\right)}\right)\right)\right)\right) \]

12. Simplified 69.9%

    \[\leadsto cosTheta\_i \cdot \left(\left(\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\color{blue}{\frac{-1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v}}{-v}}}\right) \]

13. Final simplification 69.9%

    \[\leadsto cosTheta\_i \cdot \left(\left(\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}\right) \]
14. Add Preprocessing

Alternative 6: 64.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i \cdot \left(\left(\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}}\right) \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i
  (*
   (* (/ (- 1.0 (* sinTheta_O (/ sinTheta_i v))) v) cosTheta_O)
   (/ (/ 0.5 v) (/ (+ 1.0 (/ 0.16666666666666666 (* v v))) v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * ((((1.0f - (sinTheta_O * (sinTheta_i / v))) / v) * cosTheta_O) * ((0.5f / v) / ((1.0f + (0.16666666666666666f / (v * v))) / v)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v 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 * ((((1.0e0 - (sintheta_o * (sintheta_i / v))) / v) * costheta_o) * ((0.5e0 / v) / ((1.0e0 + (0.16666666666666666e0 / (v * v))) / v)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(Float32(Float32(Float32(Float32(1.0) - Float32(sinTheta_O * Float32(sinTheta_i / v))) / v) * cosTheta_O) * Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v))) / v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * ((((single(1.0) - (sinTheta_O * (sinTheta_i / v))) / v) * cosTheta_O) * ((single(0.5) / v) / ((single(1.0) + (single(0.16666666666666666) / (v * v))) / v)));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right) \cdot \color{blue}{\frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}\right) \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. exp-neg N/A

      \[\leadsto \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot \frac{1}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-inv N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{\color{blue}{1}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   7. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   8. associate-/l* N/A

      \[\leadsto \left(cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right) \cdot \frac{\color{blue}{1}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]

   9. associate-*l* N/A

      \[\leadsto cosTheta\_i \cdot \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{\frac{cosTheta\_O}{v}}{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}\right), \color{blue}{\left(\frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)}\right)\right) \]

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \left(\frac{cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{\frac{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}{cosTheta\_O}}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(\frac{1}{2}, v\right)}, \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{v}\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\left(\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \left(\frac{v}{sinTheta\_O}\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   9. /-lowering-/.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right)\right)\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto cosTheta\_i \cdot \left(\color{blue}{\left(\frac{\frac{1}{v}}{e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}} \cdot cosTheta\_O\right)} \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

7. Taylor expanded in v around inf

   \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{\left(\frac{1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{v}\right)}, cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + -1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\frac{1}{2}}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \left(\frac{sinTheta\_i}{v}\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f32 99.0%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto cosTheta\_i \cdot \left(\left(\color{blue}{\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v}} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \]

10. Taylor expanded in v around inf

    \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right), \color{blue}{v}\right)\right)\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right), v\right)\right)\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6} \cdot 1}{{v}^{2}}\right)\right), v\right)\right)\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{\frac{1}{6}}{{v}^{2}}\right)\right), v\right)\right)\right)\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left({v}^{2}\right)\right)\right), v\right)\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \left(v \cdot v\right)\right)\right), v\right)\right)\right)\right) \]

    7. *-lowering-*.f32 63.7%

       \[\leadsto \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(sinTheta\_O, \mathsf{/.f32}\left(sinTheta\_i, v\right)\right)\right), v\right), cosTheta\_O\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(v, v\right)\right)\right), v\right)\right)\right)\right) \]

12. Simplified 63.7%

    \[\leadsto cosTheta\_i \cdot \left(\left(\frac{1 - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{v} \cdot cosTheta\_O\right) \cdot \frac{\frac{0.5}{v}}{\color{blue}{\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v}}}\right) \]
13. Add Preprocessing

Alternative 7: 68.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{\frac{0.5}{v}}{\frac{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ 0.5 v)
  (/
   (/
    (-
     (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
     -1.0)
    v)
   (/ (* cosTheta_i cosTheta_O) v))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) / (((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) - -1.0f) / v) / ((cosTheta_i * cosTheta_O) / v));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) / (((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v)) - (-1.0e0)) / v) / ((costheta_i * costheta_o) / v))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) - Float32(-1.0)) / v) / Float32(Float32(cosTheta_i * cosTheta_O) / v)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) / (((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v)) - single(-1.0)) / v) / ((cosTheta_i * cosTheta_O) / v));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{v \cdot 2}} \]

   4. clear-num N/A

      \[\leadsto \frac{\frac{1}{\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}}{\color{blue}{v} \cdot 2} \]

   5. associate-/l/ N/A

      \[\leadsto \frac{1}{\color{blue}{\left(v \cdot 2\right) \cdot \frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{v \cdot 2}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{v \cdot 2}\right), \color{blue}{\left(\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2 \cdot v}\right), \left(\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2}\right), v\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{\sinh \color{blue}{\left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(e^{\color{blue}{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)\right) \]

4. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}}}} \]

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \color{blue}{v}\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 93.3%

   \[\leadsto \frac{\frac{0.5}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v}}}} \]

8. Taylor expanded in v around -inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), v\right)\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)}, v\right)\right)\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)}, v\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right), \left(\mathsf{neg}\left(v\right)\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right)}, v\right)\right)\right) \]

10. Simplified 68.1%

    \[\leadsto \frac{\frac{0.5}{v}}{\frac{\color{blue}{\frac{-1 + \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v}}{-v}}}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}} \]

11. Final simplification 68.1%

    \[\leadsto \frac{\frac{0.5}{v}}{\frac{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} - -1}{v}}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}} \]
12. Add Preprocessing

Alternative 8: 63.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{\frac{0.5}{v}}{\frac{1}{cosTheta\_i \cdot cosTheta\_O} + \frac{0.16666666666666666}{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(v \cdot v\right)\right)}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (/ 0.5 v)
  (+
   (/ 1.0 (* cosTheta_i cosTheta_O))
   (/ 0.16666666666666666 (* cosTheta_O (* cosTheta_i (* v v)))))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) / ((1.0f / (cosTheta_i * cosTheta_O)) + (0.16666666666666666f / (cosTheta_O * (cosTheta_i * (v * v)))));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) / ((1.0e0 / (costheta_i * costheta_o)) + (0.16666666666666666e0 / (costheta_o * (costheta_i * (v * v)))))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) / Float32(Float32(Float32(1.0) / Float32(cosTheta_i * cosTheta_O)) + Float32(Float32(0.16666666666666666) / Float32(cosTheta_O * Float32(cosTheta_i * Float32(v * v))))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) / ((single(1.0) / (cosTheta_i * cosTheta_O)) + (single(0.16666666666666666) / (cosTheta_O * (cosTheta_i * (v * v)))));
end

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{v \cdot 2}} \]

   4. clear-num N/A

      \[\leadsto \frac{\frac{1}{\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}}{\color{blue}{v} \cdot 2} \]

   5. associate-/l/ N/A

      \[\leadsto \frac{1}{\color{blue}{\left(v \cdot 2\right) \cdot \frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{v \cdot 2}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{v \cdot 2}\right), \color{blue}{\left(\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2 \cdot v}\right), \left(\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2}\right), v\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{\sinh \color{blue}{\left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(e^{\color{blue}{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)\right) \]

4. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}}}} \]

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \color{blue}{v}\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 93.3%

   \[\leadsto \frac{\frac{0.5}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v}}}} \]

8. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(\frac{1}{cosTheta\_O \cdot cosTheta\_i} + \frac{1}{6} \cdot \frac{1}{cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{2}\right)}\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{1}{cosTheta\_O \cdot cosTheta\_i} + \frac{\frac{1}{6} \cdot 1}{\color{blue}{cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{2}\right)}}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{1}{cosTheta\_O \cdot cosTheta\_i} + \frac{\frac{1}{6}}{\color{blue}{cosTheta\_O} \cdot \left(cosTheta\_i \cdot {v}^{2}\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\left(\frac{1}{cosTheta\_O \cdot cosTheta\_i}\right), \color{blue}{\left(\frac{\frac{1}{6}}{cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{2}\right)}\right)}\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \left(cosTheta\_O \cdot cosTheta\_i\right)\right), \left(\frac{\color{blue}{\frac{1}{6}}}{cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{2}\right)}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \left(\frac{\frac{1}{6}}{cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{2}\right)}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \color{blue}{\left(cosTheta\_O \cdot \left(cosTheta\_i \cdot {v}^{2}\right)\right)}\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(cosTheta\_i \cdot {v}^{2}\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left({v}^{2}\right)}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \left(v \cdot \color{blue}{v}\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 63.0%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right)\right), \mathsf{/.f32}\left(\frac{1}{6}, \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{*.f32}\left(v, \color{blue}{v}\right)\right)\right)\right)\right)\right) \]

10. Simplified 63.0%

    \[\leadsto \frac{\frac{0.5}{v}}{\color{blue}{\frac{1}{cosTheta\_O \cdot cosTheta\_i} + \frac{0.16666666666666666}{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(v \cdot v\right)\right)}}} \]

11. Final simplification 63.0%

    \[\leadsto \frac{\frac{0.5}{v}}{\frac{1}{cosTheta\_i \cdot cosTheta\_O} + \frac{0.16666666666666666}{cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(v \cdot v\right)\right)}} \]
12. Add Preprocessing

Alternative 9: 59.2% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{\left(2 \cdot v\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \color{blue}{2}\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right)}}{\color{blue}{v \cdot 2}} \]

   4. clear-num N/A

      \[\leadsto \frac{\frac{1}{\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}}{\color{blue}{v} \cdot 2} \]

   5. associate-/l/ N/A

      \[\leadsto \frac{1}{\color{blue}{\left(v \cdot 2\right) \cdot \frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{v \cdot 2}}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{v \cdot 2}\right), \color{blue}{\left(\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2 \cdot v}\right), \left(\frac{\sinh \left(\frac{1}{v}\right)}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2}\right), v\right), \left(\frac{\color{blue}{\sinh \left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(\frac{\sinh \color{blue}{\left(\frac{1}{v}\right)}}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\sinh \left(\frac{1}{v}\right), \color{blue}{\left(e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)}\right)\right) \]

   13. sinh-lowering-sinh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\left(\frac{1}{v}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(\mathsf{sinh.f32}\left(\mathsf{/.f32}\left(1, v\right)\right), \left(e^{\color{blue}{\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)\right) \]

4. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{v}}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot e^{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}}}}} \]

5. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(\frac{1}{cosTheta\_O \cdot cosTheta\_i}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)\right) \]

   2. *-lowering-*.f32 58.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{cosTheta\_i}\right)\right)\right) \]

7. Simplified 58.8%

   \[\leadsto \frac{\frac{0.5}{v}}{\color{blue}{\frac{1}{cosTheta\_O \cdot cosTheta\_i}}} \]

8. Final simplification 58.8%

   \[\leadsto \frac{\frac{0.5}{v}}{\frac{1}{cosTheta\_i \cdot cosTheta\_O}} \]
9. Add Preprocessing

Alternative 10: 59.0% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \color{blue}{\frac{1}{2}} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\left(\frac{cosTheta\_i}{v}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   6. /-lowering-/.f32 57.6%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \frac{1}{2}\right)\right) \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot 0.5\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot \color{blue}{\frac{1}{2}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{1}{2} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}} \cdot \frac{1}{2} \]

   4. associate-*l/ N/A

      \[\leadsto \frac{1 \cdot \frac{1}{2}}{\color{blue}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{v}}{cosTheta\_O \cdot cosTheta\_i}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{v}{cosTheta\_O \cdot cosTheta\_i}\right)}\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \left(cosTheta\_i \cdot \color{blue}{cosTheta\_O}\right)\right)\right) \]

   9. *-lowering-*.f32 58.5%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{cosTheta\_O}\right)\right)\right) \]

7. Applied egg-rr 58.5%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]
8. Add Preprocessing

Alternative 11: 58.6% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \color{blue}{\frac{1}{2}} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\left(\frac{cosTheta\_i}{v}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   6. /-lowering-/.f32 57.6%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \frac{1}{2}\right)\right) \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot 0.5\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot \color{blue}{\frac{1}{2}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \frac{1}{2} \]

   3. div-inv N/A

      \[\leadsto \left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{v}\right) \cdot \frac{1}{2} \]

   4. associate-*l* N/A

      \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot \frac{1}{2}\right)} \]

   5. associate-/r/ N/A

      \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{\color{blue}{\frac{v}{\frac{1}{2}}}} \]

   6. clear-num N/A

      \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\frac{1}{2}}{\color{blue}{v}} \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \left(\frac{\color{blue}{\frac{1}{2}}}{v}\right)\right) \]

   10. /-lowering-/.f32 57.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right) \]

7. Applied egg-rr 57.7%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{0.5}{v}} \]

8. Final simplification 57.7%

   \[\leadsto \frac{0.5}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]
9. Add Preprocessing

Alternative 12: 58.6% accurate, 31.4× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v 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 (/ 0.5 v))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v 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 * (0.5e0 / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} \cdot \color{blue}{\frac{1}{2}} \]

   2. associate-/l* N/A

      \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right) \cdot \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot \frac{1}{2}\right)}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\left(\frac{cosTheta\_i}{v}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   6. /-lowering-/.f32 57.6%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(\mathsf{/.f32}\left(cosTheta\_i, v\right), \frac{1}{2}\right)\right) \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} \cdot 0.5\right)} \]
6. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i \cdot \frac{1}{2}}{\color{blue}{v}}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \left(cosTheta\_i \cdot \color{blue}{\frac{\frac{1}{2}}{v}}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)\right) \]

   4. /-lowering-/.f32 57.6%

      \[\leadsto \mathsf{*.f32}\left(cosTheta\_O, \mathsf{*.f32}\left(cosTheta\_i, \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{v}\right)\right)\right) \]

7. Applied egg-rr 57.6%

   \[\leadsto cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{0.5}{v}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))