HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 21.5s

Alternatives: 16

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: 98.5% 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

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

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
   (/
    (* (/ 1.0 v) (/ cosTheta_O_m (* (* v 2.0) (sinh (/ 1.0 v)))))
    (/ (pow (exp sinTheta_i) (/ sinTheta_O 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 * (((1.0f / v) * (cosTheta_O_m / ((v * 2.0f) * sinhf((1.0f / v))))) / (powf(expf(sinTheta_i), (sinTheta_O / 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 * (((1.0e0 / v) * (costheta_o_m / ((v * 2.0e0) * sinh((1.0e0 / v))))) / ((exp(sintheta_i) ** (sintheta_o / 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(Float32(Float32(1.0) / v) * Float32(cosTheta_O_m / Float32(Float32(v * Float32(2.0)) * sinh(Float32(Float32(1.0) / v))))) / Float32((exp(sinTheta_i) ^ Float32(sinTheta_O / 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(1.0) / v) * (cosTheta_O_m / ((v * single(2.0)) * sinh((single(1.0) / v))))) / ((exp(sinTheta_i) ^ (sinTheta_O / 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* 92.9%

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

   2. *-commutative 92.9%

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

   3. distribute-neg-frac 92.9%

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

   4. distribute-rgt-neg-out 92.9%

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

   5. associate-/l* 92.9%

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

   6. *-commutative 92.9%

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

   7. *-commutative 92.9%

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

   8. associate-/l* 92.9%

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

3. Simplified 92.9%

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

   1. expm1-log1p-u 92.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}}\right)\right)} \]

   2. expm1-udef 48.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}}\right)} - 1} \]

6. Applied egg-rr 48.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(e^{\frac{sinTheta\_i}{v}}\right)}^{\left(-sinTheta\_O\right)}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)\right)} - 1} \]

8. Simplified 98.4%

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

   1. associate-*r/ 98.4%

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

   2. associate-/l/ 98.5%

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

   3. associate-*r* 98.5%

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

10. Applied egg-rr 98.5%

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

    1. associate-/l* 98.5%

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

    2. *-commutative 98.5%

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

12. Simplified 98.5%

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

    1. *-un-lft-identity 98.5%

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

    2. times-frac 98.6%

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

14. Applied egg-rr 98.6%

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

15. Final simplification 98.6%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

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
   (/
    (*
     (/ (pow (exp (/ (- sinTheta_i) v)) sinTheta_O) (* v 2.0))
     (* cosTheta_i_m (/ cosTheta_O_m v)))
    (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 * (((powf(expf((-sinTheta_i / v)), sinTheta_O) / (v * 2.0f)) * (cosTheta_i_m * (cosTheta_O_m / v))) / 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 * ((((exp((-sintheta_i / v)) ** sintheta_o) / (v * 2.0e0)) * (costheta_i_m * (costheta_o_m / v))) / 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((exp(Float32(Float32(-sinTheta_i) / v)) ^ sinTheta_O) / Float32(v * Float32(2.0))) * Float32(cosTheta_i_m * Float32(cosTheta_O_m / v))) / 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 * ((((exp((-sinTheta_i / v)) ^ sinTheta_O) / (v * single(2.0))) * (cosTheta_i_m * (cosTheta_O_m / v))) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 3: 98.7% accurate, 0.7× speedup

Math

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

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 cosTheta_i_m) v)
     (/ (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 * cosTheta_i_m) / v) * (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 * costheta_i_m) / v) * ((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 * cosTheta_i_m) / v) * 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 * cosTheta_i_m) / v) * ((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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in cosTheta_O around 0 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 98.5% accurate, 0.7× speedup

Math

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

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 (* (* v 2.0) (sinh (/ 1.0 v)))))
    (/ (pow (exp sinTheta_i) (/ sinTheta_O 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 * ((cosTheta_O_m / (v * ((v * 2.0f) * sinhf((1.0f / v))))) / (powf(expf(sinTheta_i), (sinTheta_O / 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 * ((costheta_o_m / (v * ((v * 2.0e0) * sinh((1.0e0 / v))))) / ((exp(sintheta_i) ** (sintheta_o / 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(cosTheta_O_m / Float32(v * Float32(Float32(v * Float32(2.0)) * sinh(Float32(Float32(1.0) / v))))) / Float32((exp(sinTheta_i) ^ Float32(sinTheta_O / 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 * ((cosTheta_O_m / (v * ((v * single(2.0)) * sinh((single(1.0) / v))))) / ((exp(sinTheta_i) ^ (sinTheta_O / 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* 92.9%

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

   2. *-commutative 92.9%

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

   3. distribute-neg-frac 92.9%

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

   4. distribute-rgt-neg-out 92.9%

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

   5. associate-/l* 92.9%

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

   6. *-commutative 92.9%

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

   7. *-commutative 92.9%

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

   8. associate-/l* 92.9%

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

3. Simplified 92.9%

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

   1. expm1-log1p-u 92.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}}\right)\right)} \]

   2. expm1-udef 48.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\frac{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}}}\right)} - 1} \]

6. Applied egg-rr 48.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(e^{\frac{sinTheta\_i}{v}}\right)}^{\left(-sinTheta\_O\right)}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)\right)} - 1} \]

8. Simplified 98.4%

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

   1. associate-*r/ 98.4%

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

   2. associate-/l/ 98.5%

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

   3. associate-*r* 98.5%

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

10. Applied egg-rr 98.5%

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

    1. associate-/l* 98.5%

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

    2. *-commutative 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

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(sinTheta_i / v) * Float32(-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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in cosTheta_O around 0 98.4%

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

   1. associate-*r/ 98.4%

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

7. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   2. expm1-udef 48.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right)}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

   3. associate-/l* 48.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{cosTheta\_O}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i}{v}}}}\right)} - 1\right) \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

9. Applied egg-rr 48.9%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{cosTheta\_O}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i}{v}}}\right)} - 1\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]
10. Step-by-step derivation

    1. expm1-def 98.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{cosTheta\_O}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{cosTheta\_i}{v}}}\right)\right)} \cdot \frac{e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}}{v \cdot 2} \]

    2. expm1-log1p 98.3%

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

    3. associate-/r/ 98.3%

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 6: 98.5% accurate, 1.0× speedup

Math

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

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))
    (/ cosTheta_i_m (* (sinh (/ 1.0 v)) (/ v cosTheta_O_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)) * (cosTheta_i_m / (sinhf((1.0f / v)) * (v / cosTheta_O_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)) * (costheta_i_m / (sinh((1.0e0 / v)) * (v / costheta_o_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(sinTheta_i / v) * Float32(-sinTheta_O))) / Float32(v * Float32(2.0))) * Float32(cosTheta_i_m / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v / cosTheta_O_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))) * (cosTheta_i_m / (sinh((single(1.0) / v)) * (v / cosTheta_O_m)))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

   1. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.3%

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

   2. *-commutative 98.3%

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

   3. associate-*r/ 98.3%

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

   4. *-commutative 98.3%

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

   5. associate-/r/ 98.3%

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

   6. frac-times 98.1%

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

   7. *-un-lft-identity 98.1%

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

8. Applied egg-rr 98.1%

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

9. Final simplification 98.1%

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

Alternative 7: 98.5% accurate, 1.0× speedup

Math

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

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_i_m (/ cosTheta_O_m 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_i_m * (cosTheta_O_m / 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_i_m * (costheta_o_m / 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_i_m * Float32(cosTheta_O_m / v)) / sinh(Float32(Float32(1.0) / v))) * Float32(exp(Float32(Float32(sinTheta_i / v) * Float32(-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_i_m * (cosTheta_O_m / 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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 8: 98.5% accurate, 1.0× speedup

Math

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

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 cosTheta_i_m) v)
     (exp (- (/ (* sinTheta_i sinTheta_O) v))))
    (* 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 * cosTheta_i_m) / v) * expf(-((sinTheta_i * sinTheta_O) / v))) / (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 * costheta_i_m) / v) * exp(-((sintheta_i * sintheta_o) / v))) / (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 * cosTheta_i_m) / v) * exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v)))) / Float32(v * Float32(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 * cosTheta_i_m) / v) * exp(-((sinTheta_i * sinTheta_O) / v))) / (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. Add Preprocessing

3. Final simplification 98.5%

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

Alternative 9: 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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around inf 97.6%

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

   1. associate-/l* 97.6%

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

9. Simplified 97.6%

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

10. Final simplification 97.6%

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

Alternative 10: 98.3% accurate, 1.0× speedup

Math

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

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
   (/
    (* 0.5 (/ (* cosTheta_O_m cosTheta_i_m) (pow 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 * ((0.5f * ((cosTheta_O_m * cosTheta_i_m) / powf(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 * ((0.5e0 * ((costheta_o_m * costheta_i_m) / (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(0.5) * Float32(Float32(cosTheta_O_m * cosTheta_i_m) / (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 * ((single(0.5) * ((cosTheta_O_m * cosTheta_i_m) / (v ^ single(2.0)))) / sinh((single(1.0) / v))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

   1. associate-*l/ 98.5%

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

   2. exp-prod 98.5%

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

6. Applied egg-rr 98.5%

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

7. Taylor expanded in v around inf 97.6%

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

8. Final simplification 97.6%

   \[\leadsto \frac{0.5 \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2}}}{\sinh \left(\frac{1}{v}\right)} \]
9. Add Preprocessing

Alternative 11: 58.4% accurate, 24.4× speedup

Math

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

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_i_m (/ 1.0 (/ v cosTheta_O_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_i_m * (1.0f / (v / cosTheta_O_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_i_m * (1.0e0 / (v / costheta_o_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_i_m * Float32(Float32(1.0) / Float32(v / cosTheta_O_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_i_m * (single(1.0) / (v / cosTheta_O_m)))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in v around inf 54.7%

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

   1. associate-*r/ 54.7%

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

7. Simplified 54.7%

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

   1. *-commutative 54.7%

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

   2. associate-/r/ 54.7%

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

   3. clear-num 55.3%

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

9. Applied egg-rr 55.3%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{v}{cosTheta\_O}}{cosTheta\_i}}} \]
10. Step-by-step derivation

    1. associate-/r/ 54.7%

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

11. Simplified 54.7%

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

12. Final simplification 54.7%

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

Alternative 12: 58.9% 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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in v around inf 54.7%

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

   1. associate-*r/ 54.7%

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

7. Simplified 54.7%

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

   1. *-commutative 54.7%

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

   2. associate-/r/ 54.7%

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

9. Applied egg-rr 54.7%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. associate-/l* 54.7%

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

    2. associate-*r/ 54.7%

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

    3. *-commutative 54.7%

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

11. Applied egg-rr 54.7%

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

    1. *-commutative 54.7%

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

    2. associate-*r/ 54.7%

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

    3. clear-num 55.3%

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

    4. *-commutative 55.3%

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

13. Applied egg-rr 55.3%

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

14. Final simplification 55.3%

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

Alternative 13: 58.9% accurate, 24.4× speedup

Math

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

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(Float32(v / 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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in v around inf 54.7%

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

   1. associate-*r/ 54.7%

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

7. Simplified 54.7%

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

   1. *-commutative 54.7%

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

   2. associate-/r/ 54.7%

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

   3. clear-num 55.3%

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

9. Applied egg-rr 55.3%

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

10. Final simplification 55.3%

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

Alternative 14: 58.4% 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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in v around inf 54.7%

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

   1. associate-*r/ 54.7%

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

7. Simplified 54.7%

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

8. Final simplification 54.7%

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

Alternative 15: 58.4% accurate, 31.4× speedup

Math

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

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_i_m (/ cosTheta_O_m v)) 0.5))))

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_i_m * (cosTheta_O_m / v)) * 0.5f));
}

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_i_m * (costheta_o_m / v)) * 0.5e0))
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_i_m * Float32(cosTheta_O_m / v)) * Float32(0.5))))
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_i_m * (cosTheta_O_m / v)) * single(0.5)));
end

Derivation

1. Initial program 98.5%

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

   1. *-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 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in v around inf 54.7%

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

   1. associate-*r/ 54.7%

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

7. Simplified 54.7%

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

   1. *-commutative 54.7%

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

   2. associate-/r/ 54.7%

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

9. Applied egg-rr 54.7%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{cosTheta\_i}{\frac{v}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. associate-/l* 54.7%

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

    2. associate-*r/ 54.7%

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

    3. *-commutative 54.7%

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

11. Applied egg-rr 54.7%

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

12. Final simplification 54.7%

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

Alternative 16: 58.4% accurate, 31.4× speedup

Math

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

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_i_m (/ v cosTheta_O_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_i_m / (v / cosTheta_O_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_i_m / (v / costheta_o_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_i_m / Float32(v / cosTheta_O_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_i_m / (v / cosTheta_O_m))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.4%

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

   4. *-commutative 98.4%

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

   5. associate-*l/ 98.3%

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

   6. distribute-neg-frac 98.3%

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

   7. distribute-lft-neg-out 98.3%

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

   8. associate-*l/ 98.3%

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

   9. *-commutative 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in v around inf 54.7%

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

   1. associate-*r/ 54.7%

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

7. Simplified 54.7%

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

   1. *-commutative 54.7%

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

   2. associate-/r/ 54.7%

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

9. Applied egg-rr 54.7%

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

10. Final simplification 54.7%

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

Reproduce

herbie shell --seed 2024041 
(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)))