HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 17.2s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Initial Program: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Alternative 1: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left({\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \left(cosTheta\_i\_m \cdot \left(\frac{1}{v} \cdot \frac{1}{v}\right)\right)}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (pow (exp sinTheta_i) (/ sinTheta_O (- v)))
   (/
    (* cosTheta_O (* cosTheta_i_m (* (/ 1.0 v) (/ 1.0 v))))
    (* (sinh (/ 1.0 v)) 2.0)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (powf(expf(sinTheta_i), (sinTheta_O / -v)) * ((cosTheta_O * (cosTheta_i_m * ((1.0f / v) * (1.0f / v)))) / (sinhf((1.0f / v)) * 2.0f)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((exp(sintheta_i) ** (sintheta_o / -v)) * ((costheta_o * (costheta_i_m * ((1.0e0 / v) * (1.0e0 / v)))) / (sinh((1.0e0 / v)) * 2.0e0)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32((exp(sinTheta_i) ^ Float32(sinTheta_O / Float32(-v))) * Float32(Float32(cosTheta_O * Float32(cosTheta_i_m * Float32(Float32(Float32(1.0) / v) * Float32(Float32(1.0) / v)))) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp(sinTheta_i) ^ (sinTheta_O / -v)) * ((cosTheta_O * (cosTheta_i_m * ((single(1.0) / v) * (single(1.0) / v)))) / (sinh((single(1.0) / v)) * single(2.0))));
end

Derivation

1. Initial program 98.5%

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

   1. times-frac 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.4%

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

   4. distribute-frac-neg2 98.4%

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

   5. associate-/l* 98.4%

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

   6. exp-prod 98.4%

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

   7. *-commutative 98.4%

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

   8. associate-/l* 98.4%

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

   9. associate-/l* 98.5%

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

3. Simplified 98.5%

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

   1. div-inv 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. times-frac 98.7%

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

6. Applied egg-rr 98.7%

   \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(\frac{cosTheta\_i}{1} \cdot \frac{\frac{1}{v}}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
7. Step-by-step derivation

   1. /-rgt-identity 98.7%

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

   2. associate-*r/ 98.7%

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

8. Applied egg-rr 98.7%

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

   1. associate-/l* 98.7%

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

10. Simplified 98.7%

    \[\leadsto {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{-v}\right)} \cdot \frac{cosTheta\_O \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{\frac{1}{v}}{v}\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]
11. Step-by-step derivation

    1. div-inv 98.9%

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

12. Applied egg-rr 98.9%

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

13. Final simplification 98.9%

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

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{cosTheta\_O \cdot cosTheta\_i\_m}{\left(\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right) \cdot {\left(e^{sinTheta\_i}\right)}^{\left(\frac{sinTheta\_O}{v}\right)}} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (* cosTheta_O cosTheta_i_m)
   (*
    (* (sinh (/ 1.0 v)) (* v (* v 2.0)))
    (pow (exp sinTheta_i) (/ sinTheta_O v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((cosTheta_O * cosTheta_i_m) / ((sinhf((1.0f / v)) * (v * (v * 2.0f))) * powf(expf(sinTheta_i), (sinTheta_O / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((costheta_o * costheta_i_m) / ((sinh((1.0e0 / v)) * (v * (v * 2.0e0))) * (exp(sintheta_i) ** (sintheta_o / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(cosTheta_O * cosTheta_i_m) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0)))) * (exp(sinTheta_i) ^ Float32(sinTheta_O / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((cosTheta_O * cosTheta_i_m) / ((sinh((single(1.0) / v)) * (v * (v * single(2.0)))) * (exp(sinTheta_i) ^ (sinTheta_O / v))));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\_m\right)\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (*
    (exp (/ (* sinTheta_i sinTheta_O) (- v)))
    (* (/ 1.0 v) (* cosTheta_O cosTheta_i_m)))
   (* v (* (sinh (/ 1.0 v)) 2.0)))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((expf(((sinTheta_i * sinTheta_O) / -v)) * ((1.0f / v) * (cosTheta_O * cosTheta_i_m))) / (v * (sinhf((1.0f / v)) * 2.0f)));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((exp(((sintheta_i * sintheta_o) / -v)) * ((1.0e0 / v) * (costheta_o * costheta_i_m))) / (v * (sinh((1.0e0 / v)) * 2.0e0)))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_O * cosTheta_i_m))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((single(1.0) / v) * (cosTheta_O * cosTheta_i_m))) / (v * (sinh((single(1.0) / v)) * single(2.0))));
end

Derivation

1. Initial program 98.5%

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

   1. div-inv 98.6%

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

4. Applied egg-rr 98.6%

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

5. Final simplification 98.6%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(\frac{\frac{2}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta\_i\_m \cdot \frac{cosTheta\_O}{v}}{2}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/ (/ 2.0 v) (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))
   (/ (* cosTheta_i_m (/ cosTheta_O v)) 2.0))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((2.0f / v) / (expf((1.0f / v)) - expf((-1.0f / v)))) * ((cosTheta_i_m * (cosTheta_O / v)) / 2.0f));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((2.0e0 / v) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))) * ((costheta_i_m * (costheta_o / v)) / 2.0e0))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(2.0) / v) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v)))) * Float32(Float32(cosTheta_i_m * Float32(cosTheta_O / v)) / Float32(2.0))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((single(2.0) / v) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))) * ((cosTheta_i_m * (cosTheta_O / v)) / single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ 98.5%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.4%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. times-frac 98.4%

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

   3. add-sqr-sqrt 65.7%

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

   4. add-sqr-sqrt 98.4%

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

   5. associate-*r/ 98.4%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.2%

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

   8. sqr-neg 98.2%

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

   9. sqrt-unprod 98.2%

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

   10. add-sqr-sqrt 98.2%

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

   11. associate-/l* 98.2%

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

   12. pow-exp 98.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in sinTheta_i around 0 98.4%

   \[\leadsto \color{blue}{\frac{2}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]
8. Step-by-step derivation

   1. associate-/r* 98.6%

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

   2. rec-exp 98.6%

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

   3. distribute-neg-frac 98.6%

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

   4. metadata-eval 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

    \[\leadsto \frac{\frac{2}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]
11. Add Preprocessing

Alternative 5: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(\frac{cosTheta\_i\_m \cdot \frac{cosTheta\_O}{v}}{2} \cdot \frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   (/ (* cosTheta_i_m (/ cosTheta_O v)) 2.0)
   (/ (/ 1.0 v) (sinh (/ 1.0 v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (((cosTheta_i_m * (cosTheta_O / v)) / 2.0f) * ((1.0f / v) / sinhf((1.0f / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((costheta_i_m * (costheta_o / v)) / 2.0e0) * ((1.0e0 / v) / sinh((1.0e0 / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_i_m * Float32(cosTheta_O / v)) / Float32(2.0)) * Float32(Float32(Float32(1.0) / v) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (((cosTheta_i_m * (cosTheta_O / v)) / single(2.0)) * ((single(1.0) / v) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ 98.5%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.4%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. times-frac 98.4%

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

   3. add-sqr-sqrt 65.7%

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

   4. add-sqr-sqrt 98.4%

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

   5. associate-*r/ 98.4%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.2%

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

   8. sqr-neg 98.2%

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

   9. sqrt-unprod 98.2%

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

   10. add-sqr-sqrt 98.2%

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

   11. associate-/l* 98.2%

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

   12. pow-exp 98.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in sinTheta_i around 0 98.4%

   \[\leadsto \color{blue}{\frac{2}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]
8. Step-by-step derivation

   1. *-un-lft-identity 98.4%

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

   2. associate-/r* 98.6%

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

   3. rec-exp 98.6%

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

   4. sinh-undef 98.5%

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

   5. *-commutative 98.5%

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

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left(1 \cdot \frac{\frac{2}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2}\right)} \cdot \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \]
10. Step-by-step derivation

    1. *-lft-identity 98.5%

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

    2. associate-/r* 98.2%

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

    3. metadata-eval 98.2%

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

    4. associate-*l/ 98.2%

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

    5. associate-/r/ 98.2%

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

    6. associate-/l* 98.2%

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

    7. associate-/r* 98.5%

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

    8. associate-/l* 98.5%

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

    9. metadata-eval 98.5%

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

    10. *-commutative 98.5%

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

    11. *-lft-identity 98.5%

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

11. Simplified 98.5%

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

12. Final simplification 98.5%

    \[\leadsto \frac{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}{2} \cdot \frac{\frac{1}{v}}{\sinh \left(\frac{1}{v}\right)} \]
13. Add Preprocessing

Alternative 6: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot {\left(\frac{2}{cosTheta\_i\_m \cdot \frac{cosTheta\_O}{v}}\right)}^{-1} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (pow (/ 2.0 (* cosTheta_i_m (/ cosTheta_O v))) -1.0)))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * powf((2.0f / (cosTheta_i_m * (cosTheta_O / v))), -1.0f);
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((2.0e0 / (costheta_i_m * (costheta_o / v))) ** (-1.0e0))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * (Float32(Float32(2.0) / Float32(cosTheta_i_m * Float32(cosTheta_O / v))) ^ Float32(-1.0)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((single(2.0) / (cosTheta_i_m * (cosTheta_O / v))) ^ single(-1.0));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ 98.5%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.4%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. times-frac 98.4%

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

   3. add-sqr-sqrt 65.7%

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

   4. add-sqr-sqrt 98.4%

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

   5. associate-*r/ 98.4%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.2%

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

   8. sqr-neg 98.2%

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

   9. sqrt-unprod 98.2%

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

   10. add-sqr-sqrt 98.2%

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

   11. associate-/l* 98.2%

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

   12. pow-exp 98.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in v around inf 58.2%

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

   1. clear-num 58.5%

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

   2. inv-pow 58.5%

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

9. Applied egg-rr 58.5%

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

10. Final simplification 58.5%

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

Alternative 7: 58.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_i\_m \cdot \frac{\frac{cosTheta\_O}{v}}{2}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_i_m (/ (/ cosTheta_O v) 2.0))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_i_m * ((cosTheta_O / v) / 2.0f));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (costheta_i_m * ((costheta_o / v) / 2.0e0))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_i_m * Float32(Float32(cosTheta_O / v) / Float32(2.0))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_i_m * ((cosTheta_O / v) / single(2.0)));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ 98.5%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.4%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. times-frac 98.4%

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

   3. add-sqr-sqrt 65.7%

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

   4. add-sqr-sqrt 98.4%

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

   5. associate-*r/ 98.4%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.2%

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

   8. sqr-neg 98.2%

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

   9. sqrt-unprod 98.2%

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

   10. add-sqr-sqrt 98.2%

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

   11. associate-/l* 98.2%

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

   12. pow-exp 98.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in v around inf 58.2%

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

   1. associate-/l* 58.2%

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

9. Applied egg-rr 58.2%

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

10. Final simplification 58.2%

    \[\leadsto cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{2} \]
11. Add Preprocessing

Alternative 8: 58.0% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{cosTheta\_O \cdot \frac{cosTheta\_i\_m}{v}}{2} \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (/ (* cosTheta_O (/ cosTheta_i_m v)) 2.0)))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * ((cosTheta_O * (cosTheta_i_m / v)) / 2.0f);
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((costheta_o * (costheta_i_m / v)) / 2.0e0)
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(Float32(cosTheta_O * Float32(cosTheta_i_m / v)) / Float32(2.0)))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((cosTheta_O * (cosTheta_i_m / v)) / single(2.0));
end

Derivation

1. Initial program 98.5%

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

   1. associate-*r/ 98.5%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.4%

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

   9. associate-*r/ 98.5%

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

3. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. times-frac 98.4%

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

   3. add-sqr-sqrt 65.7%

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

   4. add-sqr-sqrt 98.4%

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

   5. associate-*r/ 98.4%

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

   6. add-sqr-sqrt -0.0%

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

   7. sqrt-unprod 98.2%

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

   8. sqr-neg 98.2%

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

   9. sqrt-unprod 98.2%

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

   10. add-sqr-sqrt 98.2%

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

   11. associate-/l* 98.2%

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

   12. pow-exp 98.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in v around inf 58.2%

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

8. Taylor expanded in cosTheta_i around 0 58.2%

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

   1. associate-/l* 58.2%

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

10. Simplified 58.2%

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

11. Final simplification 58.2%

    \[\leadsto \frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{2} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))