HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 17.7s

Alternatives: 13

Speedup: 0.7×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (((costheta_o_m * ((exp((-sintheta_i / v)) ** sintheta_o) / (v * 2.0e0))) / sinh((1.0e0 / v))) * (costheta_i / v))
end function

Julia

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_O_m * Float32((exp(Float32(Float32(-sinTheta_i) / v)) ^ sinTheta_O) / Float32(v * Float32(2.0)))) / sinh(Float32(Float32(1.0) / v))) * Float32(cosTheta_i / v)))
end

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta_O around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-/r* 98.4%

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

   3. associate-*l/ 98.4%

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

   4. associate-/r/ 98.6%

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

   5. rec-exp 98.6%

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

   6. sinh-def 98.6%

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

   7. associate-*l/ 98.7%

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

   8. associate-*r/ 98.6%

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

   9. associate-/l* 98.4%

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

7. Simplified 98.4%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

9. Applied egg-rr 98.5%

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

    1. associate-/r/ 98.8%

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((exp(((-sintheta_i / v) * sintheta_o)) / (v * 2.0e0)) * (costheta_i / (sinh((1.0e0 / v)) * (v / costheta_o_m))))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((exp(((-sinTheta_i / v) * sinTheta_O)) / (v * single(2.0))) * (cosTheta_i / (sinh((single(1.0) / v)) * (v / cosTheta_O_m))));
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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. *-commutative 59.0%

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

   2. associate-*r/ 59.0%

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

   3. associate-/l* 59.0%

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

6. Applied egg-rr 98.7%

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

   1. expm1-log1p-u 98.7%

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

   2. expm1-udef 53.9%

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

   3. associate-/l/ 53.9%

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

8. Applied egg-rr 53.9%

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

   1. expm1-def 98.6%

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

   2. expm1-log1p 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (((costheta_i / (v / costheta_o_m)) / sinh((1.0e0 / v))) * (exp(((-sintheta_i / v) * sintheta_o)) / (v * 2.0e0)))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (((cosTheta_i / (v / cosTheta_O_m)) / sinh((single(1.0) / v))) * (exp(((-sinTheta_i / v) * sinTheta_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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

   1. *-commutative 59.0%

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

   2. associate-*r/ 59.0%

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

   3. associate-/l* 59.0%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * ((cosTheta_i / v) * ((cosTheta_O_m / v) / (expf((1.0f / v)) - expf((-1.0f / v)))));
}

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_i / v) * ((costheta_o_m / v) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_i / v) * ((cosTheta_O_m / v) / (exp((single(1.0) / v)) - exp((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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta_O around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-/r* 98.4%

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

   3. associate-*l/ 98.4%

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

   4. associate-/r/ 98.6%

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

   5. rec-exp 98.6%

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

   6. sinh-def 98.6%

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

   7. associate-*l/ 98.7%

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

   8. associate-*r/ 98.6%

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

   9. associate-/l* 98.4%

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

7. Simplified 98.4%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

9. Applied egg-rr 98.5%

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

    1. associate-/r/ 98.8%

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

11. Simplified 98.8%

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

12. Taylor expanded in cosTheta_O around 0 98.6%

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

    1. *-commutative 98.6%

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

    2. times-frac 98.7%

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

    3. rec-exp 98.7%

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

    4. distribute-neg-frac 98.7%

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

    5. metadata-eval 98.7%

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

    6. neg-mul-1 98.7%

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

    7. associate-/l* 98.7%

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

    8. distribute-neg-frac 98.7%

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

14. Simplified 98.7%

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

15. Taylor expanded in sinTheta_O around 0 98.5%

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

    1. associate-/r* 98.5%

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

17. Simplified 98.5%

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

18. Final simplification 98.5%

    \[\leadsto \frac{cosTheta_i}{v} \cdot \frac{\frac{cosTheta_O}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \]
19. Add Preprocessing

Alternative 5: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * ((cosTheta_i / v) * (cosTheta_O_m / ((v * 2.0f) * sinhf((1.0f / v)))));
}

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_i / v) * (costheta_o_m / ((v * 2.0e0) * sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_i / v) * (cosTheta_O_m / ((v * single(2.0)) * 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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta_O around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-/r* 98.4%

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

   3. associate-*l/ 98.4%

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

   4. associate-/r/ 98.6%

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

   5. rec-exp 98.6%

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

   6. sinh-def 98.6%

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

   7. associate-*l/ 98.7%

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

   8. associate-*r/ 98.6%

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

   9. associate-/l* 98.4%

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

7. Simplified 98.4%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

9. Applied egg-rr 98.5%

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

    1. associate-/r/ 98.8%

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

11. Simplified 98.8%

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

12. Taylor expanded in sinTheta_i around 0 98.5%

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

    1. rec-exp 98.5%

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

    2. distribute-neg-frac 98.5%

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

    3. metadata-eval 98.5%

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

14. Simplified 98.5%

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

    1. expm1-log1p-u 98.5%

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

    2. expm1-udef 62.2%

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

    3. associate-/r* 62.2%

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

    4. div-inv 62.2%

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

    5. neg-mul-1 62.2%

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

    6. sinh-undef 62.2%

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

16. Applied egg-rr 62.2%

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

    1. expm1-def 98.5%

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

    2. expm1-log1p 98.5%

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

    3. associate-/l/ 98.5%

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

    4. *-commutative 98.5%

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

    5. associate-*r* 98.5%

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

    6. *-commutative 98.5%

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

18. Simplified 98.5%

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

19. Final simplification 98.5%

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

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* (/ cosTheta_i v) (/ (/ (* cosTheta_O_m 0.5) v) (sinh (/ 1.0 v))))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_i / v) * (((costheta_o_m * 0.5e0) / v) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_i / v) * (((cosTheta_O_m * single(0.5)) / 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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in cosTheta_O around 0 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-/r* 98.4%

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

   3. associate-*l/ 98.4%

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

   4. associate-/r/ 98.6%

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

   5. rec-exp 98.6%

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

   6. sinh-def 98.6%

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

   7. associate-*l/ 98.7%

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

   8. associate-*r/ 98.6%

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

   9. associate-/l* 98.4%

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

7. Simplified 98.4%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

9. Applied egg-rr 98.5%

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

    1. associate-/r/ 98.8%

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

11. Simplified 98.8%

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

12. Taylor expanded in sinTheta_i around 0 98.5%

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

    1. associate-*r/ 98.5%

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

    2. *-commutative 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 7: 58.9% accurate, 24.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O_m}}{cosTheta_i}}\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (/ 1.0 (/ (/ v cosTheta_O_m) cosTheta_i)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 * (1.0e0 / ((v / costheta_o_m) / costheta_i)))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) * (single(1.0) / ((v / cosTheta_O_m) / cosTheta_i)));
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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.0%

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

   1. associate-*l/ 59.0%

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

7. Applied egg-rr 59.0%

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

   1. *-commutative 59.0%

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

   2. associate-*r/ 59.0%

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

   3. associate-/l* 59.0%

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

   4. clear-num 59.4%

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

9. Applied egg-rr 59.4%

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

10. Final simplification 59.4%

    \[\leadsto 0.5 \cdot \frac{1}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \]
11. Add Preprocessing

Alternative 8: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(0.5 \cdot \left(cosTheta_i \cdot \frac{cosTheta_O_m}{v}\right)\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (* cosTheta_i (/ cosTheta_O_m v)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 * (costheta_i * (costheta_o_m / v)))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) * (cosTheta_i * (cosTheta_O_m / 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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.0%

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

   1. associate-*l/ 59.0%

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

7. Applied egg-rr 59.0%

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

8. Final simplification 59.0%

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

Alternative 9: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(0.5 \cdot \frac{cosTheta_O_m}{\frac{v}{cosTheta_i}}\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (/ cosTheta_O_m (/ v cosTheta_i)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 * (costheta_o_m / (v / costheta_i)))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) * (cosTheta_O_m / (v / cosTheta_i)));
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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.0%

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

   1. associate-*l/ 59.0%

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

7. Applied egg-rr 59.0%

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

8. Taylor expanded in cosTheta_O around 0 59.0%

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

   1. associate-/l* 59.0%

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

10. Simplified 59.0%

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

11. Final simplification 59.0%

    \[\leadsto 0.5 \cdot \frac{cosTheta_O}{\frac{v}{cosTheta_i}} \]
12. Add Preprocessing

Alternative 10: 58.4% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(\frac{cosTheta_i}{\frac{v}{cosTheta_O_m}} \cdot 0.5\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* (/ cosTheta_i (/ v cosTheta_O_m)) 0.5)))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_i / (v / costheta_o_m)) * 0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.0%

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

   1. associate-*l/ 59.0%

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

7. Applied egg-rr 59.0%

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

   1. *-commutative 59.0%

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

   2. associate-*r/ 59.0%

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

   3. associate-/l* 59.0%

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

9. Applied egg-rr 59.0%

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

10. Final simplification 59.0%

    \[\leadsto \frac{cosTheta_i}{\frac{v}{cosTheta_O}} \cdot 0.5 \]
11. Add Preprocessing

Alternative 11: 58.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(0.5 \cdot \frac{cosTheta_O_m \cdot cosTheta_i}{v}\right) \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* 0.5 (/ (* cosTheta_O_m cosTheta_i) v))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 * ((costheta_o_m * costheta_i) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.0%

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

6. Final simplification 59.0%

   \[\leadsto 0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{v} \]
7. Add Preprocessing

Alternative 12: 58.9% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \frac{0.5}{\frac{v}{cosTheta_O_m \cdot cosTheta_i}} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (/ 0.5 (/ v (* cosTheta_O_m cosTheta_i)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 / (v / (costheta_o_m * costheta_i)))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) / (v / (cosTheta_O_m * cosTheta_i)));
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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.0%

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

   1. associate-*r/ 59.0%

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

   2. associate-/l* 59.4%

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

7. Simplified 59.4%

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

8. Final simplification 59.4%

   \[\leadsto \frac{0.5}{\frac{v}{cosTheta_O \cdot cosTheta_i}} \]
9. Add Preprocessing

Alternative 13: 58.9% accurate, 31.4× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \frac{0.5}{\frac{\frac{v}{cosTheta_O_m}}{cosTheta_i}} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (/ 0.5 (/ (/ v cosTheta_O_m) cosTheta_i))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 / ((v / costheta_o_m) / costheta_i))
end function

Julia

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

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (single(0.5) / ((v / cosTheta_O_m) / cosTheta_i));
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. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.7%

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

   4. *-commutative 98.7%

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

   5. associate-*l/ 98.6%

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

   6. distribute-neg-frac 98.6%

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

   7. distribute-lft-neg-out 98.6%

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

   8. associate-*l/ 98.6%

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

   9. *-commutative 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in v around inf 59.0%

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

   1. associate-*l/ 59.0%

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

7. Applied egg-rr 59.0%

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

8. Taylor expanded in cosTheta_O around 0 59.0%

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

   1. associate-/l* 59.0%

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

10. Simplified 59.0%

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

    1. associate-*r/ 59.0%

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

12. Applied egg-rr 59.0%

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

    1. associate-/l* 59.4%

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

    2. associate-/l/ 59.4%

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

    3. *-commutative 59.4%

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

    4. associate-/l/ 59.4%

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

14. Simplified 59.4%

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

15. Final simplification 59.4%

    \[\leadsto \frac{0.5}{\frac{\frac{v}{cosTheta_O}}{cosTheta_i}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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)))