HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 20.3s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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/ 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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. div-inv 98.6%

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

6. Applied egg-rr 98.6%

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

7. Final simplification 98.6%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. div-inv 98.7%

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

   2. *-commutative 98.7%

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   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 cosTheta_O around 0 98.5%

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

   1. associate-*r/ 98.6%

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

7. Simplified 98.6%

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

   1. div-inv 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.6%

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

    2. *-rgt-identity 98.6%

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

    3. associate-*r/ 98.5%

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

    4. *-commutative 98.5%

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

    5. associate-*r/ 98.5%

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

    6. associate-*r/ 98.6%

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

11. Simplified 98.6%

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

12. Final simplification 98.6%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   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 cosTheta_O around 0 98.5%

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

   1. associate-*r/ 98.6%

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

7. Simplified 98.6%

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

   1. associate-*r/ 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. clear-num 96.6%

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

9. Applied egg-rr 96.6%

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

    1. associate-/r/ 98.7%

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

11. Simplified 98.7%

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

    1. associate-*l/ 98.5%

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

    2. *-un-lft-identity 98.5%

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

    3. associate-*l/ 98.5%

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

    4. *-commutative 98.5%

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

    5. associate-*r/ 98.6%

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

    6. *-commutative 98.6%

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

    7. associate-/l/ 98.7%

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

13. Applied egg-rr 98.7%

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

14. Final simplification 98.7%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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 cosTheta_O around 0 98.5%

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

   1. associate-*r/ 98.6%

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

7. Simplified 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 7: 98.3% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (* (/ cosTheta_O (pow v 2.0)) (* cosTheta_i_m (/ 0.5 (sinh (/ 1.0 v)))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * ((costheta_o / (v ** 2.0e0)) * (costheta_i_m * (0.5e0 / sinh((1.0e0 / v)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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 sinTheta_i around 0 98.4%

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

   1. times-frac 98.3%

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

   2. rec-exp 98.4%

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

   3. distribute-neg-frac 98.4%

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

   4. metadata-eval 98.4%

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

7. Simplified 98.4%

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

   1. div-inv 98.3%

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

   2. div-inv 98.3%

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

   3. neg-mul-1 98.3%

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

   4. sinh-undef 98.3%

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

9. Applied egg-rr 98.3%

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

    1. associate-/r* 98.3%

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

    2. metadata-eval 98.3%

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

11. Simplified 98.3%

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

12. Final simplification 98.3%

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

Alternative 8: 64.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}} \cdot \left(-cosTheta\_i\_m\right)}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{cosTheta\_O}} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (* (exp (- (/ sinTheta_O (/ v sinTheta_i)))) (- cosTheta_i_m))
   (/ (- (* v -2.0) (/ 0.3333333333333333 v)) cosTheta_O))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * ((exp(-(sintheta_o / (v / sintheta_i))) * -costheta_i_m) / (((v * (-2.0e0)) - (0.3333333333333333e0 / v)) / costheta_o))
end function

Julia

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

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * ((exp(-(sinTheta_O / (v / sinTheta_i))) * -cosTheta_i_m) / (((v * single(-2.0)) - (single(0.3333333333333333) / v)) / cosTheta_O));
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

5. Taylor expanded in v around inf 64.4%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}}} \]

6. Taylor expanded in cosTheta_i around -inf 64.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]
7. Step-by-step derivation

   1. mul-1-neg 64.7%

      \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]

   2. mul-1-neg 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}} \]

   3. associate-/l* 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}} \]

   4. associate-*r/ 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta\_O \cdot v}}} \]

   5. metadata-eval 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333}}{cosTheta\_O \cdot v}} \]

8. Simplified 64.7%

   \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{cosTheta\_O \cdot v}}} \]

9. Taylor expanded in cosTheta_O around 0 64.7%

   \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{\color{blue}{\frac{-2 \cdot v - 0.3333333333333333 \cdot \frac{1}{v}}{cosTheta\_O}}} \]
10. Step-by-step derivation

    1. *-commutative 64.7%

       \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{\frac{\color{blue}{v \cdot -2} - 0.3333333333333333 \cdot \frac{1}{v}}{cosTheta\_O}} \]

    2. associate-*r/ 64.7%

       \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{\frac{v \cdot -2 - \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}}{cosTheta\_O}} \]

    3. metadata-eval 64.7%

       \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{\frac{v \cdot -2 - \frac{\color{blue}{0.3333333333333333}}{v}}{cosTheta\_O}} \]

11. Simplified 64.7%

    \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{\color{blue}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{cosTheta\_O}}} \]

12. Final simplification 64.7%

    \[\leadsto \frac{e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}} \cdot \left(-cosTheta\_i\right)}{\frac{v \cdot -2 - \frac{0.3333333333333333}{v}}{cosTheta\_O}} \]
13. Add Preprocessing

Alternative 9: 64.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := -2 \cdot \frac{v}{cosTheta\_O} + 0.3333333333333333 \cdot \frac{-1}{cosTheta\_O \cdot v}\\ cosTheta\_i\_s \cdot \left(\frac{cosTheta\_i\_m \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v \cdot t\_0} - \frac{cosTheta\_i\_m}{t\_0}\right) \end{array} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (+
          (* -2.0 (/ v cosTheta_O))
          (* 0.3333333333333333 (/ -1.0 (* cosTheta_O v))))))
   (*
    cosTheta_i_s
    (-
     (/ (* cosTheta_i_m (* sinTheta_i sinTheta_O)) (* v t_0))
     (/ cosTheta_i_m t_0)))))

C

cosTheta_i_m = fabs(cosTheta_i);
cosTheta_i_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (-2.0f * (v / cosTheta_O)) + (0.3333333333333333f * (-1.0f / (cosTheta_O * v)));
	return cosTheta_i_s * (((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / (v * t_0)) - (cosTheta_i_m / t_0));
}

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    t_0 = ((-2.0e0) * (v / costheta_o)) + (0.3333333333333333e0 * ((-1.0e0) / (costheta_o * v)))
    code = costheta_i_s * (((costheta_i_m * (sintheta_i * sintheta_o)) / (v * t_0)) - (costheta_i_m / t_0))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0, cosTheta_i)
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(Float32(-2.0) * Float32(v / cosTheta_O)) + Float32(Float32(0.3333333333333333) * Float32(Float32(-1.0) / Float32(cosTheta_O * v))))
	return Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_i_m * Float32(sinTheta_i * sinTheta_O)) / Float32(v * t_0)) - Float32(cosTheta_i_m / t_0)))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (single(-2.0) * (v / cosTheta_O)) + (single(0.3333333333333333) * (single(-1.0) / (cosTheta_O * v)));
	tmp = cosTheta_i_s * (((cosTheta_i_m * (sinTheta_i * sinTheta_O)) / (v * t_0)) - (cosTheta_i_m / t_0));
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

5. Taylor expanded in v around inf 64.4%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}}} \]

6. Taylor expanded in cosTheta_i around -inf 64.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]
7. Step-by-step derivation

   1. mul-1-neg 64.7%

      \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]

   2. mul-1-neg 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}} \]

   3. associate-/l* 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}} \]

   4. associate-*r/ 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta\_O \cdot v}}} \]

   5. metadata-eval 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333}}{cosTheta\_O \cdot v}} \]

8. Simplified 64.7%

   \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{cosTheta\_O \cdot v}}} \]

9. Taylor expanded in sinTheta_O around 0 64.7%

   \[\leadsto -\color{blue}{\left(-1 \cdot \frac{cosTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v \cdot \left(-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}\right)} + \frac{cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}\right)} \]

10. Final simplification 64.7%

    \[\leadsto \frac{cosTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}{v \cdot \left(-2 \cdot \frac{v}{cosTheta\_O} + 0.3333333333333333 \cdot \frac{-1}{cosTheta\_O \cdot v}\right)} - \frac{cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} + 0.3333333333333333 \cdot \frac{-1}{cosTheta\_O \cdot v}} \]
11. Add Preprocessing

Alternative 10: 64.0% accurate, 15.7× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ cosTheta_i_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \frac{-cosTheta\_i\_m}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{cosTheta\_O \cdot v}} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i_s cosTheta_i_m cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (/
   (- cosTheta_i_m)
   (- (* -2.0 (/ v cosTheta_O)) (/ 0.3333333333333333 (* cosTheta_O v))))))

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (-costheta_i_m / (((-2.0e0) * (v / costheta_o)) - (0.3333333333333333e0 / (costheta_o * v))))
end function

Julia

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

MATLAB

cosTheta_i_m = abs(cosTheta_i);
cosTheta_i_s = sign(double(cosTheta_i)) * abs(1.0);
cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i_s, cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (-cosTheta_i_m / ((single(-2.0) * (v / cosTheta_O)) - (single(0.3333333333333333) / (cosTheta_O * v))));
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* 94.9%

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

   2. *-commutative 94.9%

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

   3. distribute-neg-frac 94.9%

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

   4. distribute-rgt-neg-out 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

3. Simplified 94.9%

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

5. Taylor expanded in v around inf 64.4%

   \[\leadsto \frac{e^{\frac{sinTheta\_i}{\frac{v}{-sinTheta\_O}}}}{\color{blue}{2 \cdot \frac{v}{cosTheta\_O \cdot cosTheta\_i} + 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)}}} \]

6. Taylor expanded in cosTheta_i around -inf 64.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]
7. Step-by-step derivation

   1. mul-1-neg 64.7%

      \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]

   2. mul-1-neg 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{\color{blue}{-\frac{sinTheta\_O \cdot sinTheta\_i}{v}}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}} \]

   3. associate-/l* 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\color{blue}{\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}} \]

   4. associate-*r/ 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta\_O \cdot v}}} \]

   5. metadata-eval 64.7%

      \[\leadsto -\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333}}{cosTheta\_O \cdot v}} \]

8. Simplified 64.7%

   \[\leadsto \color{blue}{-\frac{cosTheta\_i \cdot e^{-\frac{sinTheta\_O}{\frac{v}{sinTheta\_i}}}}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{cosTheta\_O \cdot v}}} \]

9. Taylor expanded in sinTheta_O around 0 64.7%

   \[\leadsto -\color{blue}{\frac{cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} - 0.3333333333333333 \cdot \frac{1}{cosTheta\_O \cdot v}}} \]
10. Step-by-step derivation

    1. associate-*r/ 64.7%

       \[\leadsto -\frac{cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{cosTheta\_O \cdot v}}} \]

    2. metadata-eval 64.7%

       \[\leadsto -\frac{cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{\color{blue}{0.3333333333333333}}{cosTheta\_O \cdot v}} \]

11. Simplified 64.7%

    \[\leadsto -\color{blue}{\frac{cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{cosTheta\_O \cdot v}}} \]

12. Final simplification 64.7%

    \[\leadsto \frac{-cosTheta\_i}{-2 \cdot \frac{v}{cosTheta\_O} - \frac{0.3333333333333333}{cosTheta\_O \cdot v}} \]
13. Add Preprocessing

Alternative 11: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 * (costheta_o * (costheta_i_m / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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 59.4%

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

   1. associate-*r/ 59.4%

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

7. Simplified 59.4%

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

8. Final simplification 59.4%

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

Alternative 12: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 * (costheta_i_m / (v / costheta_o)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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 59.4%

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

   1. associate-*r/ 59.4%

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

7. Simplified 59.4%

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

   1. associate-*r/ 59.4%

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

   2. associate-*l/ 59.4%

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

   3. *-commutative 59.4%

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

   4. clear-num 59.4%

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

   5. un-div-inv 59.4%

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

9. Applied egg-rr 59.4%

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

10. Final simplification 59.4%

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

Alternative 13: 58.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 * ((costheta_o * costheta_i_m) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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 59.4%

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

6. Final simplification 59.4%

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

Alternative 14: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

cosTheta_i_m = abs(cosTheta_i)
cosTheta_i_s = copysign(1.0d0, cosTheta_i)
NOTE: cosTheta_i_m, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i_s, costheta_i_m, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_i_s * (0.5e0 / (v / (costheta_o * costheta_i_m)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*l/ 98.5%

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

   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 59.4%

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

   1. associate-*r/ 59.5%

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

   2. associate-/l* 60.0%

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

7. Simplified 60.0%

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

8. Final simplification 60.0%

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

Reproduce

herbie shell --seed 2024039 
(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)))