HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 23.6s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. div-inv 98.5%

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

   4. times-frac 98.7%

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

6. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*r/ 98.7%

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

8. Applied egg-rr 98.7%

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

9. Final simplification 98.7%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

6. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*r/ 98.7%

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

8. Applied egg-rr 98.7%

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

   1. associate-*l/ 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. times-frac 98.7%

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

   4. add-sqr-sqrt 46.2%

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

   5. sqrt-unprod 98.5%

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

   6. sqr-neg 98.5%

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

   7. sqrt-unprod 52.3%

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

   8. add-sqr-sqrt 98.2%

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

   9. times-frac 98.2%

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

   10. *-un-lft-identity 98.2%

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

10. Applied egg-rr 98.2%

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

11. Final simplification 98.2%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. times-frac 98.7%

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

   4. add-sqr-sqrt 46.2%

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

   5. sqrt-unprod 98.5%

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

   6. sqr-neg 98.5%

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

   7. sqrt-unprod 52.3%

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

   8. add-sqr-sqrt 98.2%

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

   9. times-frac 98.2%

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

   10. *-un-lft-identity 98.2%

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

8. Applied egg-rr 98.0%

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

   1. expm1-log1p-u 98.1%

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

   2. associate-*l/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-/l* 98.4%

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

10. Applied egg-rr 98.4%

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

11. Final simplification 98.4%

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

Alternative 5: 64.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta_O_s \cdot \frac{0.5 \cdot \left(\left(cosTheta_i \cdot cosTheta_O_m\right) \cdot e^{\frac{-sinTheta_O}{\frac{v}{sinTheta_i}}}\right)}{v + \frac{0.16666666666666666}{v}} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (*
    0.5
    (* (* cosTheta_i cosTheta_O_m) (exp (/ (- sinTheta_O) (/ v sinTheta_i)))))
   (+ v (/ 0.16666666666666666 v)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((0.5e0 * ((costheta_i * costheta_o_m) * exp((-sintheta_o / (v / sintheta_i))))) / (v + (0.16666666666666666e0 / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 93.4%

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

   2. *-commutative 93.4%

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

   3. distribute-neg-frac 93.4%

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

   4. distribute-rgt-neg-out 93.4%

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

   5. associate-/l* 93.4%

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

   6. *-commutative 93.4%

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

   7. *-commutative 93.4%

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

   8. associate-/l* 93.4%

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

3. Simplified 93.4%

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

   1. associate-*r* 93.4%

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

   2. associate-/l* 93.4%

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

   3. *-commutative 93.4%

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

   4. associate-*l/ 93.4%

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

   5. times-frac 93.4%

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

6. Applied egg-rr 93.4%

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

   1. associate-/l* 93.4%

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

   2. associate-/r/ 93.3%

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

   3. associate-/r/ 93.5%

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

8. Simplified 93.5%

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

9. Taylor expanded in v around inf 63.7%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\color{blue}{\left(0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}\right)} \cdot \frac{2}{cosTheta_i}} \]

10. Taylor expanded in cosTheta_O around 0 64.6%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_O \cdot \left(cosTheta_i \cdot e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)}{v + 0.16666666666666666 \cdot \frac{1}{v}}} \]
11. Step-by-step derivation

    1. associate-*r/ 64.6%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(cosTheta_O \cdot \left(cosTheta_i \cdot e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)\right)}{v + 0.16666666666666666 \cdot \frac{1}{v}}} \]

    2. associate-*r* 64.6%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot e^{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}}\right)}}{v + 0.16666666666666666 \cdot \frac{1}{v}} \]

    3. mul-1-neg 64.6%

       \[\leadsto \frac{0.5 \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot e^{\color{blue}{-\frac{sinTheta_O \cdot sinTheta_i}{v}}}\right)}{v + 0.16666666666666666 \cdot \frac{1}{v}} \]

    4. associate-/l* 64.6%

       \[\leadsto \frac{0.5 \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot e^{-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}}\right)}{v + 0.16666666666666666 \cdot \frac{1}{v}} \]

    5. associate-*r/ 64.6%

       \[\leadsto \frac{0.5 \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot e^{-\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)}{v + \color{blue}{\frac{0.16666666666666666 \cdot 1}{v}}} \]

    6. metadata-eval 64.6%

       \[\leadsto \frac{0.5 \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot e^{-\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)}{v + \frac{\color{blue}{0.16666666666666666}}{v}} \]

12. Simplified 64.6%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(cosTheta_O \cdot cosTheta_i\right) \cdot e^{-\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right)}{v + \frac{0.16666666666666666}{v}}} \]

13. Final simplification 64.6%

    \[\leadsto \frac{0.5 \cdot \left(\left(cosTheta_i \cdot cosTheta_O\right) \cdot e^{\frac{-sinTheta_O}{\frac{v}{sinTheta_i}}}\right)}{v + \frac{0.16666666666666666}{v}} \]
14. Add Preprocessing

Alternative 6: 64.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} cosTheta_O_m = \left|cosTheta_O\right| \\ cosTheta_O_s = \mathsf{copysign}\left(1, cosTheta_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot \frac{1}{v \cdot cosTheta_O_m} + \frac{v}{cosTheta_O_m}\\ cosTheta_O_s \cdot \left(-0.5 \cdot \frac{\left(sinTheta_i \cdot sinTheta_O\right) \cdot cosTheta_i}{v \cdot t_0} + 0.5 \cdot \frac{cosTheta_i}{t_0}\right) \end{array} \end{array} \]

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (+
          (* 0.16666666666666666 (/ 1.0 (* v cosTheta_O_m)))
          (/ v cosTheta_O_m))))
   (*
    cosTheta_O_s
    (+
     (* -0.5 (/ (* (* sinTheta_i sinTheta_O) cosTheta_i) (* v t_0)))
     (* 0.5 (/ cosTheta_i t_0))))))

C

cosTheta_O_m = fabs(cosTheta_O);
cosTheta_O_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (0.16666666666666666f * (1.0f / (v * cosTheta_O_m))) + (v / cosTheta_O_m);
	return cosTheta_O_s * ((-0.5f * (((sinTheta_i * sinTheta_O) * cosTheta_i) / (v * t_0))) + (0.5f * (cosTheta_i / t_0)));
}

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = (0.16666666666666666e0 * (1.0e0 / (v * costheta_o_m))) + (v / costheta_o_m)
    code = costheta_o_s * (((-0.5e0) * (((sintheta_i * sintheta_o) * costheta_i) / (v * t_0))) + (0.5e0 * (costheta_i / t_0)))
end function

Julia

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(v * cosTheta_O_m))) + Float32(v / cosTheta_O_m))
	return Float32(cosTheta_O_s * Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(sinTheta_i * sinTheta_O) * cosTheta_i) / Float32(v * t_0))) + Float32(Float32(0.5) * Float32(cosTheta_i / t_0))))
end

MATLAB

cosTheta_O_m = abs(cosTheta_O);
cosTheta_O_s = sign(double(cosTheta_O)) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	t_0 = (single(0.16666666666666666) * (single(1.0) / (v * cosTheta_O_m))) + (v / cosTheta_O_m);
	tmp = cosTheta_O_s * ((single(-0.5) * (((sinTheta_i * sinTheta_O) * cosTheta_i) / (v * t_0))) + (single(0.5) * (cosTheta_i / t_0)));
end

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 93.4%

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

   2. *-commutative 93.4%

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

   3. distribute-neg-frac 93.4%

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

   4. distribute-rgt-neg-out 93.4%

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

   5. associate-/l* 93.4%

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

   6. *-commutative 93.4%

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

   7. *-commutative 93.4%

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

   8. associate-/l* 93.4%

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

3. Simplified 93.4%

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

   1. associate-*r* 93.4%

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

   2. associate-/l* 93.4%

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

   3. *-commutative 93.4%

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

   4. associate-*l/ 93.4%

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

   5. times-frac 93.4%

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

6. Applied egg-rr 93.4%

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

   1. associate-/l* 93.4%

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

   2. associate-/r/ 93.3%

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

   3. associate-/r/ 93.5%

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

8. Simplified 93.5%

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

9. Taylor expanded in v around inf 63.7%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\color{blue}{\left(0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}\right)} \cdot \frac{2}{cosTheta_i}} \]

10. Taylor expanded in sinTheta_i around 0 64.6%

    \[\leadsto \color{blue}{-0.5 \cdot \frac{cosTheta_i \cdot \left(sinTheta_O \cdot sinTheta_i\right)}{v \cdot \left(0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}\right)} + 0.5 \cdot \frac{cosTheta_i}{0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}}} \]

11. Final simplification 64.6%

    \[\leadsto -0.5 \cdot \frac{\left(sinTheta_i \cdot sinTheta_O\right) \cdot cosTheta_i}{v \cdot \left(0.16666666666666666 \cdot \frac{1}{v \cdot cosTheta_O} + \frac{v}{cosTheta_O}\right)} + 0.5 \cdot \frac{cosTheta_i}{0.16666666666666666 \cdot \frac{1}{v \cdot cosTheta_O} + \frac{v}{cosTheta_O}} \]
12. Add Preprocessing

Alternative 7: 64.6% accurate, 16.9× speedup

Math

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

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   0.5
   (/
    cosTheta_i
    (+ (/ v cosTheta_O_m) (/ (/ 0.16666666666666666 v) cosTheta_O_m))))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (0.5e0 * (costheta_i / ((v / costheta_o_m) + ((0.16666666666666666e0 / v) / costheta_o_m))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 93.4%

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

   2. *-commutative 93.4%

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

   3. distribute-neg-frac 93.4%

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

   4. distribute-rgt-neg-out 93.4%

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

   5. associate-/l* 93.4%

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

   6. *-commutative 93.4%

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

   7. *-commutative 93.4%

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

   8. associate-/l* 93.4%

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

3. Simplified 93.4%

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

   1. associate-*r* 93.4%

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

   2. associate-/l* 93.4%

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

   3. *-commutative 93.4%

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

   4. associate-*l/ 93.4%

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

   5. times-frac 93.4%

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

6. Applied egg-rr 93.4%

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

   1. associate-/l* 93.4%

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

   2. associate-/r/ 93.3%

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

   3. associate-/r/ 93.5%

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

8. Simplified 93.5%

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

9. Taylor expanded in v around inf 63.7%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\color{blue}{\left(0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}\right)} \cdot \frac{2}{cosTheta_i}} \]

10. Taylor expanded in sinTheta_i around 0 64.6%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}}} \]
11. Step-by-step derivation

    1. +-commutative 64.6%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\color{blue}{\frac{v}{cosTheta_O} + 0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v}}} \]

    2. associate-*r/ 64.6%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \color{blue}{\frac{0.16666666666666666 \cdot 1}{cosTheta_O \cdot v}}} \]

    3. metadata-eval 64.6%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \frac{\color{blue}{0.16666666666666666}}{cosTheta_O \cdot v}} \]

    4. associate-/l/ 64.6%

       \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \color{blue}{\frac{\frac{0.16666666666666666}{v}}{cosTheta_O}}} \]

12. Simplified 64.6%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \frac{\frac{0.16666666666666666}{v}}{cosTheta_O}}} \]

13. Final simplification 64.6%

    \[\leadsto 0.5 \cdot \frac{cosTheta_i}{\frac{v}{cosTheta_O} + \frac{\frac{0.16666666666666666}{v}}{cosTheta_O}} \]
14. Add Preprocessing

Alternative 8: 64.6% accurate, 16.9× speedup

Math

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

FPCore

cosTheta_O_m = (fabs.f32 cosTheta_O)
cosTheta_O_s = (copysign.f32 1 cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (* cosTheta_i 0.5)
   (+ (/ v cosTheta_O_m) (/ (/ 0.16666666666666666 v) cosTheta_O_m)))))

C

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

Fortran

cosTheta_O_m = abs(cosTheta_O)
cosTheta_O_s = copysign(1.0d0, cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_i * 0.5e0) / ((v / costheta_o_m) + ((0.16666666666666666e0 / v) / costheta_o_m)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. associate-/l* 93.4%

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

   2. *-commutative 93.4%

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

   3. distribute-neg-frac 93.4%

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

   4. distribute-rgt-neg-out 93.4%

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

   5. associate-/l* 93.4%

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

   6. *-commutative 93.4%

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

   7. *-commutative 93.4%

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

   8. associate-/l* 93.4%

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

3. Simplified 93.4%

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

   1. associate-*r* 93.4%

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

   2. associate-/l* 93.4%

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

   3. *-commutative 93.4%

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

   4. associate-*l/ 93.4%

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

   5. times-frac 93.4%

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

6. Applied egg-rr 93.4%

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

   1. associate-/l* 93.4%

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

   2. associate-/r/ 93.3%

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

   3. associate-/r/ 93.5%

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

8. Simplified 93.5%

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

9. Taylor expanded in v around inf 63.7%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\color{blue}{\left(0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}\right)} \cdot \frac{2}{cosTheta_i}} \]

10. Taylor expanded in sinTheta_i around 0 64.6%

    \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta_i}{0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}}} \]
11. Step-by-step derivation

    1. associate-*r/ 64.6%

       \[\leadsto \color{blue}{\frac{0.5 \cdot cosTheta_i}{0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}}} \]

    2. *-commutative 64.6%

       \[\leadsto \frac{\color{blue}{cosTheta_i \cdot 0.5}}{0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v} + \frac{v}{cosTheta_O}} \]

    3. +-commutative 64.6%

       \[\leadsto \frac{cosTheta_i \cdot 0.5}{\color{blue}{\frac{v}{cosTheta_O} + 0.16666666666666666 \cdot \frac{1}{cosTheta_O \cdot v}}} \]

    4. associate-*r/ 64.6%

       \[\leadsto \frac{cosTheta_i \cdot 0.5}{\frac{v}{cosTheta_O} + \color{blue}{\frac{0.16666666666666666 \cdot 1}{cosTheta_O \cdot v}}} \]

    5. metadata-eval 64.6%

       \[\leadsto \frac{cosTheta_i \cdot 0.5}{\frac{v}{cosTheta_O} + \frac{\color{blue}{0.16666666666666666}}{cosTheta_O \cdot v}} \]

    6. associate-/l/ 64.6%

       \[\leadsto \frac{cosTheta_i \cdot 0.5}{\frac{v}{cosTheta_O} + \color{blue}{\frac{\frac{0.16666666666666666}{v}}{cosTheta_O}}} \]

12. Simplified 64.6%

    \[\leadsto \color{blue}{\frac{cosTheta_i \cdot 0.5}{\frac{v}{cosTheta_O} + \frac{\frac{0.16666666666666666}{v}}{cosTheta_O}}} \]

13. Final simplification 64.6%

    \[\leadsto \frac{cosTheta_i \cdot 0.5}{\frac{v}{cosTheta_O} + \frac{\frac{0.16666666666666666}{v}}{cosTheta_O}} \]
14. Add Preprocessing

Alternative 9: 59.3% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.6%

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

   2. *-commutative 58.6%

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

7. Applied egg-rr 58.6%

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

   1. associate-*l/ 58.6%

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

   2. *-commutative 58.6%

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

   3. clear-num 59.2%

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

   4. *-commutative 59.2%

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

9. Applied egg-rr 59.2%

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

10. Final simplification 59.2%

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

Alternative 10: 58.8% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.6%

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

7. Simplified 58.6%

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

8. Final simplification 58.6%

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

Alternative 11: 58.8% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. expm1-log1p-u 98.4%

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

6. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.5%

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

   2. associate-*l/ 98.5%

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

   3. associate-*r/ 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*r/ 98.7%

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

8. Applied egg-rr 98.7%

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

9. Taylor expanded in v around inf 58.6%

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

    1. associate-/l* 58.6%

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

11. Simplified 58.6%

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

12. Final simplification 58.6%

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

Alternative 12: 58.8% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.6%

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

6. Final simplification 58.6%

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

Alternative 13: 58.8% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.6%

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

7. Simplified 58.6%

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

8. Taylor expanded in cosTheta_O around 0 58.6%

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

   1. associate-*r/ 58.6%

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

   2. *-commutative 58.6%

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

   3. associate-*l* 58.7%

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

10. Simplified 58.7%

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

11. Final simplification 58.7%

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

Alternative 14: 58.8% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

   1. associate-*l/ 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-/l* 98.5%

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

   4. associate-/r/ 98.5%

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

   5. *-un-lft-identity 98.5%

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

   6. times-frac 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.6%

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

   2. associate-*l* 58.7%

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

   3. *-commutative 58.7%

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

   4. associate-*l/ 58.7%

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

   5. associate-*r/ 58.7%

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

   6. *-commutative 58.7%

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

9. Simplified 58.7%

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

10. Final simplification 58.7%

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

Alternative 15: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   6. distribute-neg-frac 98.5%

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

   7. distribute-lft-neg-out 98.5%

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

   8. associate-*l/ 98.5%

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

   9. *-commutative 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 58.6%

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

   1. associate-*r/ 58.7%

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

7. Simplified 58.7%

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

8. Final simplification 58.7%

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

Reproduce

herbie shell --seed 2024017 
(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)))