HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 20.4s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (*
     (* (/ cosTheta_O_m v) cosTheta_i_m)
     (/ (pow (exp (/ (- sinTheta_i) v)) sinTheta_O) (* v 2.0)))
    (sinh (/ 1.0 v))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((((costheta_o_m / v) * costheta_i_m) * ((exp((-sintheta_i / v)) ** sintheta_o) / (v * 2.0e0))) / sinh((1.0e0 / v))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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. associate-*l/ 98.7%

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

   2. exp-prod 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* (* cosTheta_O_m cosTheta_i_m) (/ (/ 1.0 v) (sinh (/ 1.0 v))))
    (/ (exp (* (/ (- sinTheta_i) v) sinTheta_O)) (* v 2.0))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((costheta_o_m * costheta_i_m) * ((1.0e0 / v) / sinh((1.0e0 / v)))) * (exp(((-sintheta_i / v) * sintheta_o)) / (v * 2.0e0))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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 98.5%

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

   1. associate-/l* 98.6%

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

7. Simplified 98.6%

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

   1. associate-/l* 98.5%

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

   2. div-inv 98.6%

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

   3. *-un-lft-identity 98.6%

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

   4. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ (exp (* (/ (- sinTheta_i) v) sinTheta_O)) (* v 2.0))
    (* (/ cosTheta_O_m (sinh (/ 1.0 v))) (/ cosTheta_i_m v))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((exp(((-sintheta_i / v) * sintheta_o)) / (v * 2.0e0)) * ((costheta_o_m / sinh((1.0e0 / v))) * (costheta_i_m / v))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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 98.5%

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

   1. associate-/l* 98.6%

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

7. Simplified 98.6%

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

   1. associate-/l* 98.5%

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

   2. div-inv 98.6%

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

   3. *-un-lft-identity 98.6%

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

   4. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

    1. expm1-log1p-u 98.7%

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

    2. expm1-udef 50.9%

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

    3. /-rgt-identity 50.9%

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

    4. *-commutative 50.9%

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

    5. associate-/l/ 50.9%

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

11. Applied egg-rr 50.9%

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

    1. expm1-def 98.5%

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

    2. expm1-log1p 98.5%

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

    3. associate-*l/ 98.5%

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

    4. *-lft-identity 98.5%

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

    5. /-rgt-identity 98.5%

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

    6. times-frac 98.5%

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

    7. /-rgt-identity 98.5%

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

13. Simplified 98.5%

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

14. Final simplification 98.5%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* (/ cosTheta_O_m v) (/ 1.0 v))
    (/ cosTheta_i_m (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((costheta_o_m / v) * (1.0e0 / v)) * (costheta_i_m / (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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 sinTheta_i around 0 98.6%

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

   1. times-frac 98.5%

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

   2. rec-exp 98.5%

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

   3. distribute-neg-frac 98.5%

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

   4. metadata-eval 98.5%

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

7. Simplified 98.5%

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

   1. *-un-lft-identity 98.5%

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

   2. unpow2 98.5%

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

   3. times-frac 98.6%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ cosTheta_O_m v)
    (/ cosTheta_i_m (* v (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((costheta_o_m / v) * (costheta_i_m / (v * (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))))))
end function

Julia

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

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m / v) * (cosTheta_i_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 98.5%

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

   4. *-commutative 98.5%

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

   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. associate-*l/ 98.7%

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

   2. exp-prod 98.7%

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

6. Applied egg-rr 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.8%

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

8. Applied egg-rr 98.7%

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

   1. /-rgt-identity 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.7%

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

10. Simplified 98.7%

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

11. Taylor expanded in sinTheta_O around 0 98.6%

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

    1. rec-exp 98.6%

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

    2. distribute-neg-frac 98.6%

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

    3. metadata-eval 98.6%

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

13. Simplified 98.6%

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

14. Final simplification 98.6%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ cosTheta_O_m v)
    (/ (/ cosTheta_i_m v) (- (exp (/ 1.0 v)) (exp (/ -1.0 v))))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((costheta_o_m / v) * ((costheta_i_m / v) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v))))))
end function

Julia

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

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m / v) * ((cosTheta_i_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 98.5%

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

   4. *-commutative 98.5%

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

   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. associate-*l/ 98.7%

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

   2. exp-prod 98.7%

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

6. Applied egg-rr 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. associate-*l* 98.8%

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

   3. times-frac 98.8%

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

8. Applied egg-rr 98.7%

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

   1. /-rgt-identity 98.7%

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

   2. associate-/l* 98.7%

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

   3. associate-*r/ 98.7%

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

10. Simplified 98.7%

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

11. Taylor expanded in sinTheta_O around 0 98.6%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.5%

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

    3. distribute-neg-frac 98.5%

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

    4. metadata-eval 98.5%

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

13. Simplified 98.5%

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

14. Final simplification 98.5%

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

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \frac{0.5 \cdot \frac{cosTheta_O_m}{\frac{{v}^{2}}{cosTheta_i_m}}}{\sinh \left(\frac{1}{v}\right)}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* 0.5 (/ cosTheta_O_m (/ (pow v 2.0) cosTheta_i_m)))
    (sinh (/ 1.0 v))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((0.5f * (cosTheta_O_m / (powf(v, 2.0f) / cosTheta_i_m))) / sinhf((1.0f / v))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((0.5e0 * (costheta_o_m / ((v ** 2.0e0) / costheta_i_m))) / sinh((1.0e0 / v))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(0.5) * Float32(cosTheta_O_m / Float32((v ^ Float32(2.0)) / cosTheta_i_m))) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((single(0.5) * (cosTheta_O_m / ((v ^ single(2.0)) / cosTheta_i_m))) / 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 98.5%

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

   4. *-commutative 98.5%

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

   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. associate-*l/ 98.7%

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

   2. exp-prod 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in v around inf 98.4%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{cosTheta_O \cdot cosTheta_i}{{v}^{2}}}}{\sinh \left(\frac{1}{v}\right)} \]
8. Step-by-step derivation

   1. associate-/l* 98.5%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\frac{cosTheta_O}{\frac{{v}^{2}}{cosTheta_i}}}}{\sinh \left(\frac{1}{v}\right)} \]

9. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{cosTheta_O}{\frac{{v}^{2}}{cosTheta_i}}}}{\sinh \left(\frac{1}{v}\right)} \]

10. Final simplification 98.5%

    \[\leadsto \frac{0.5 \cdot \frac{cosTheta_O}{\frac{{v}^{2}}{cosTheta_i}}}{\sinh \left(\frac{1}{v}\right)} \]
11. Add Preprocessing

Alternative 8: 63.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta_i\right) \\ cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \left(cosTheta_i_s \cdot \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{2 \cdot \frac{v}{cosTheta_O_m \cdot cosTheta_i_m} + 0.3333333333333333 \cdot \frac{1}{cosTheta_O_m \cdot \left(v \cdot cosTheta_i_m\right)}}\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (exp (/ sinTheta_i (/ v (- sinTheta_O))))
    (+
     (* 2.0 (/ v (* cosTheta_O_m cosTheta_i_m)))
     (* 0.3333333333333333 (/ 1.0 (* cosTheta_O_m (* v cosTheta_i_m)))))))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (expf((sinTheta_i / (v / -sinTheta_O))) / ((2.0f * (v / (cosTheta_O_m * cosTheta_i_m))) + (0.3333333333333333f * (1.0f / (cosTheta_O_m * (v * cosTheta_i_m)))))));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (exp((sintheta_i / (v / -sintheta_o))) / ((2.0e0 * (v / (costheta_o_m * costheta_i_m))) + (0.3333333333333333e0 * (1.0e0 / (costheta_o_m * (v * costheta_i_m)))))))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(Float32(2.0) * Float32(v / Float32(cosTheta_O_m * cosTheta_i_m))) + Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / Float32(cosTheta_O_m * Float32(v * cosTheta_i_m))))))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (exp((sinTheta_i / (v / -sinTheta_O))) / ((single(2.0) * (v / (cosTheta_O_m * cosTheta_i_m))) + (single(0.3333333333333333) * (single(1.0) / (cosTheta_O_m * (v * cosTheta_i_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. associate-/l* 92.0%

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

   2. *-commutative 92.0%

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

   3. distribute-neg-frac 92.0%

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

   4. distribute-rgt-neg-out 92.0%

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

   5. associate-/l* 92.0%

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

   6. *-commutative 92.0%

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

   7. *-commutative 92.0%

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

   8. associate-/l* 92.0%

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

3. Simplified 92.0%

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

5. Taylor expanded in v around inf 60.3%

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

6. Final simplification 60.3%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{2 \cdot \frac{v}{cosTheta_O \cdot cosTheta_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta_O \cdot \left(v \cdot cosTheta_i\right)}} \]
7. Add Preprocessing

Alternative 9: 58.8% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ (exp (* (/ (- sinTheta_i) v) sinTheta_O)) (* v 2.0))
    (/ 1.0 (/ 1.0 (* cosTheta_O_m cosTheta_i_m)))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((exp(((-sintheta_i / v) * sintheta_o)) / (v * 2.0e0)) * (1.0e0 / (1.0e0 / (costheta_o_m * costheta_i_m)))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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. clear-num 92.2%

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

   2. inv-pow 92.2%

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

6. Applied egg-rr 92.2%

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

   1. unpow-1 92.2%

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

   2. associate-*l/ 92.2%

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

   3. associate-*r/ 92.2%

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

8. Simplified 92.2%

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

9. Taylor expanded in v around inf 56.3%

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

    1. *-commutative 56.3%

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

11. Simplified 56.3%

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

12. Final simplification 56.3%

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

Alternative 10: 58.6% accurate, 24.4× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* cosTheta_i_s (* 0.5 (/ 1.0 (/ v (* cosTheta_O_m cosTheta_i_m)))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 * (1.0e0 / (v / (costheta_o_m * costheta_i_m)))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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 55.8%

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

   1. clear-num 56.2%

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

   2. inv-pow 56.2%

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

7. Applied egg-rr 56.2%

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

   1. unpow-1 56.2%

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

9. Simplified 56.2%

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

10. Final simplification 56.2%

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

Alternative 11: 58.1% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* 0.5 (* cosTheta_O_m (/ cosTheta_i_m v))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 * (costheta_o_m * (costheta_i_m / v))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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 55.8%

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

   1. associate-*l/ 55.8%

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

   2. expm1-log1p-u 55.8%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)\right)} \]

   3. expm1-udef 50.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)} - 1\right)} \]

   4. associate-/r/ 50.4%

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{cosTheta_O}{\frac{v}{cosTheta_i}}}\right)} - 1\right) \]

7. Applied egg-rr 50.4%

   \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta_O}{\frac{v}{cosTheta_i}}\right)} - 1\right)} \]
8. Step-by-step derivation

   1. expm1-def 55.8%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta_O}{\frac{v}{cosTheta_i}}\right)\right)} \]

   2. expm1-log1p 55.8%

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

   3. associate-/l* 55.8%

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

   4. associate-*r/ 55.8%

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

9. Simplified 55.8%

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

10. Final simplification 55.8%

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

Alternative 12: 58.1% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* 0.5 (/ cosTheta_O_m (/ v cosTheta_i_m))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 * (costheta_o_m / (v / costheta_i_m))))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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 55.8%

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

   1. associate-*l/ 55.8%

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

   2. add-sqr-sqrt 39.2%

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

   3. sqrt-unprod 50.9%

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

   4. pow2 50.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left(\frac{cosTheta_O}{v} \cdot cosTheta_i\right)}^{2}}} \]

   5. associate-/r/ 50.9%

      \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(\frac{cosTheta_O}{\frac{v}{cosTheta_i}}\right)}}^{2}} \]

7. Applied egg-rr 50.9%

   \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{\left(\frac{cosTheta_O}{\frac{v}{cosTheta_i}}\right)}^{2}}} \]
8. Step-by-step derivation

   1. sqrt-pow1 55.8%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\frac{cosTheta_O}{\frac{v}{cosTheta_i}}\right)}^{\left(\frac{2}{2}\right)}} \]

   2. metadata-eval 55.8%

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

   3. pow1 55.8%

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

9. Applied egg-rr 55.8%

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

10. Final simplification 55.8%

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

Alternative 13: 58.1% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (* 0.5 (/ (* cosTheta_O_m cosTheta_i_m) v)))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (0.5e0 * ((costheta_o_m * costheta_i_m) / v)))
end function

Julia

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

MATLAB

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

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

   4. *-commutative 98.5%

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

   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 55.8%

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

6. Final simplification 55.8%

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

Alternative 14: 58.2% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (/ (* (* cosTheta_O_m cosTheta_i_m) 0.5) v))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * (((costheta_o_m * costheta_i_m) * 0.5e0) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.5%

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

   4. *-commutative 98.5%

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

   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 55.8%

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

   1. associate-*r/ 55.8%

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

7. Simplified 55.8%

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

8. Final simplification 55.8%

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

Alternative 15: 58.1% accurate, 31.4× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i_m, 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_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* cosTheta_i_s (/ (* cosTheta_O_m (* cosTheta_i_m 0.5)) v))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i_m, 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_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    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_s * ((costheta_o_m * (costheta_i_m * 0.5e0)) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

   \[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 98.5%

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

   4. *-commutative 98.5%

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

   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 98.5%

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

   1. associate-/l* 98.6%

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

7. Simplified 98.6%

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

   1. associate-/l* 98.5%

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

   2. div-inv 98.6%

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

   3. *-un-lft-identity 98.6%

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

   4. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

10. Taylor expanded in v around inf 55.8%

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

    1. associate-*l/ 55.8%

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

    2. *-commutative 55.8%

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

    3. associate-*r* 55.7%

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

12. Simplified 55.7%

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

    1. associate-*r/ 55.8%

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

14. Applied egg-rr 55.8%

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

15. Final simplification 55.8%

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

Reproduce

herbie shell --seed 2024018 
(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)))