HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 20.7s

Alternatives: 13

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: 98.8% accurate, 0.7× speedup

Math

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

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
  (*
   (* (/ 1.0 (* v 2.0)) (/ (/ cosTheta_i_m v) (sinh (/ 1.0 v))))
   (/ cosTheta_O (pow (exp (/ sinTheta_O v)) sinTheta_i)))))

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 * (((1.0f / (v * 2.0f)) * ((cosTheta_i_m / v) / sinhf((1.0f / v)))) * (cosTheta_O / powf(expf((sinTheta_O / v)), sinTheta_i)));
}

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 * (((1.0e0 / (v * 2.0e0)) * ((costheta_i_m / v) / sinh((1.0e0 / v)))) * (costheta_o / (exp((sintheta_o / v)) ** sintheta_i)))
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(1.0) / Float32(v * Float32(2.0))) * Float32(Float32(cosTheta_i_m / v) / sinh(Float32(Float32(1.0) / v)))) * Float32(cosTheta_O / (exp(Float32(sinTheta_O / v)) ^ sinTheta_i))))
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(1.0) / (v * single(2.0))) * ((cosTheta_i_m / v) / sinh((single(1.0) / v)))) * (cosTheta_O / (exp((sinTheta_O / v)) ^ sinTheta_i)));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. associate-*r/ 98.5%

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

   2. associate-/l* 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-/l/ 98.5%

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

   2. associate-/r/ 98.6%

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

   3. times-frac 98.6%

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

   4. *-commutative 98.6%

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

   5. associate-*l* 98.6%

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

   6. *-commutative 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

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_O (- sinTheta_i)) v))
    (* (/ 1.0 v) (* cosTheta_i_m cosTheta_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 * ((expf(((sinTheta_O * -sinTheta_i) / v)) * ((1.0f / v) * (cosTheta_i_m * cosTheta_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 * ((exp(((sintheta_o * -sintheta_i) / v)) * ((1.0e0 / v) * (costheta_i_m * costheta_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(exp(Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_i_m * cosTheta_O))) / 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_O * -sinTheta_i) / v)) * ((single(1.0) / v) * (cosTheta_i_m * cosTheta_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. 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%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
cosTheta_i_s = (copysign.f32 1 cosTheta_i)
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))
   (/
    (/ (exp (* (/ sinTheta_O v) (- sinTheta_i))) (* 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)) * ((expf(((sinTheta_O / v) * -sinTheta_i)) / (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_o / v) * -sintheta_i)) / (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(cosTheta_O / Float32(v / cosTheta_i_m)) * Float32(Float32(exp(Float32(Float32(sinTheta_O / v) * Float32(-sinTheta_i))) / 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_O / v) * -sinTheta_i)) / (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. associate-/l* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.7%

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

3. Simplified 92.7%

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

   1. add-exp-log 92.7%

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

   2. *-commutative 92.7%

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

   3. associate-*l* 92.7%

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

   4. *-commutative 92.7%

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

6. Applied egg-rr 92.7%

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

   1. expm1-log1p-u 92.7%

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

   2. expm1-udef 52.1%

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

8. Applied egg-rr 52.1%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*l/ 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. associate-/r* 98.7%

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

10. Simplified 98.7%

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

11. Final simplification 98.7%

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

Alternative 4: 63.6% accurate, 1.0× speedup

Math

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

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_i (/ v (- sinTheta_O))))
   (/
    (+ 2.0 (/ 0.3333333333333333 (pow v 2.0)))
    (/ 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 * (expf((sinTheta_i / (v / -sinTheta_O))) / ((2.0f + (0.3333333333333333f / powf(v, 2.0f))) / (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 * (exp((sintheta_i / (v / -sintheta_o))) / ((2.0e0 + (0.3333333333333333e0 / (v ** 2.0e0))) / (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(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / (v ^ Float32(2.0)))) / 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 * (exp((sinTheta_i / (v / -sinTheta_O))) / ((single(2.0) + (single(0.3333333333333333) / (v ^ single(2.0)))) / (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. associate-/l* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.7%

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

3. Simplified 92.7%

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

5. Taylor expanded in v around inf 61.8%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\frac{\color{blue}{2 + 0.3333333333333333 \cdot \frac{1}{{v}^{2}}}}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \]
6. Step-by-step derivation

   1. associate-*r/ 61.8%

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

   2. metadata-eval 61.8%

      \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\frac{2 + \frac{\color{blue}{0.3333333333333333}}{{v}^{2}}}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \]

7. Simplified 61.8%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\frac{\color{blue}{2 + \frac{0.3333333333333333}{{v}^{2}}}}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \]

8. Final simplification 61.8%

   \[\leadsto \frac{e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}}{\frac{2 + \frac{0.3333333333333333}{{v}^{2}}}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}}} \]
9. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

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) (* (* v 2.0) (sinh (/ 1.0 v)))))
   (exp (* (/ sinTheta_O v) sinTheta_i)))))

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) / ((v * 2.0f) * sinhf((1.0f / v))))) / expf(((sinTheta_O / v) * sinTheta_i)));
}

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) / ((v * 2.0e0) * sinh((1.0e0 / v))))) / exp(((sintheta_o / v) * sintheta_i)))
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) / Float32(Float32(v * Float32(2.0)) * sinh(Float32(Float32(1.0) / v))))) / exp(Float32(Float32(sinTheta_O / v) * sinTheta_i))))
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) / ((v * single(2.0)) * sinh((single(1.0) / v))))) / exp(((sinTheta_O / v) * sinTheta_i)));
end

Derivation

1. Initial program 98.5%

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

3. Simplified 98.6%

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

   1. associate-*r/ 98.5%

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

   2. associate-/l* 98.5%

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

6. Applied egg-rr 98.5%

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

   1. associate-/l/ 98.5%

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

   2. associate-/r/ 98.6%

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

   3. times-frac 98.6%

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

   4. *-commutative 98.6%

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

   5. associate-*l* 98.6%

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

   6. *-commutative 98.6%

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

8. Applied egg-rr 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. *-commutative 98.6%

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

   3. times-frac 98.7%

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

10. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.7%

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

    2. frac-times 98.6%

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

    3. *-un-lft-identity 98.6%

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

    4. pow-exp 98.6%

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

12. Applied egg-rr 98.6%

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

13. Final simplification 98.6%

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

Alternative 6: 63.6% accurate, 1.8× speedup

Math

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

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))))
   (+
    (* 2.0 (/ v (* cosTheta_i_m cosTheta_O)))
    (* 0.3333333333333333 (/ 1.0 (* cosTheta_O (* v 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 * (expf((sinTheta_i / (v / -sinTheta_O))) / ((2.0f * (v / (cosTheta_i_m * cosTheta_O))) + (0.3333333333333333f * (1.0f / (cosTheta_O * (v * 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 * (exp((sintheta_i / (v / -sintheta_o))) / ((2.0e0 * (v / (costheta_i_m * costheta_o))) + (0.3333333333333333e0 * (1.0e0 / (costheta_o * (v * 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(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(Float32(2.0) * Float32(v / Float32(cosTheta_i_m * cosTheta_O))) + Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / Float32(cosTheta_O * Float32(v * 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 * (exp((sinTheta_i / (v / -sinTheta_O))) / ((single(2.0) * (v / (cosTheta_i_m * cosTheta_O))) + (single(0.3333333333333333) * (single(1.0) / (cosTheta_O * (v * cosTheta_i_m))))));
end

Derivation

1. Initial program 98.5%

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

   1. associate-/l* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.7%

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

3. Simplified 92.7%

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

5. Taylor expanded in v around inf 61.7%

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

6. Final simplification 61.7%

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

Alternative 7: 59.4% accurate, 1.8× speedup

Math

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

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
  (*
   (/ 1.0 (* (/ v cosTheta_i_m) (/ 1.0 (* v cosTheta_O))))
   (/ (exp (* sinTheta_O (/ (- sinTheta_i) v))) (* 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 * ((1.0f / ((v / cosTheta_i_m) * (1.0f / (v * cosTheta_O)))) * (expf((sinTheta_O * (-sinTheta_i / v))) / (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 * ((1.0e0 / ((v / costheta_i_m) * (1.0e0 / (v * costheta_o)))) * (exp((sintheta_o * (-sintheta_i / v))) / (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(Float32(1.0) / Float32(Float32(v / cosTheta_i_m) * Float32(Float32(1.0) / Float32(v * cosTheta_O)))) * Float32(exp(Float32(sinTheta_O * Float32(Float32(-sinTheta_i) / v))) / 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 * ((single(1.0) / ((v / cosTheta_i_m) * (single(1.0) / (v * cosTheta_O)))) * (exp((sinTheta_O * (-sinTheta_i / v))) / (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. Step-by-step derivation

   1. expm1-log1p-u 98.5%

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

   2. expm1-udef 52.1%

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

6. Applied egg-rr 52.1%

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

   1. expm1-def 98.5%

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

   2. expm1-log1p 98.5%

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

   3. associate-*l/ 98.5%

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

   4. associate-/l* 98.5%

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

8. Simplified 98.5%

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

   1. clear-num 92.6%

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

   2. inv-pow 92.6%

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

   3. *-un-lft-identity 92.6%

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

   4. associate-/r/ 92.6%

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

   5. *-commutative 92.6%

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

   6. clear-num 92.5%

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

   7. div-inv 92.5%

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

   8. associate-/r/ 92.6%

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

   9. times-frac 92.6%

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

   10. clear-num 92.6%

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

10. Applied egg-rr 92.6%

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

    1. unpow-1 92.6%

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

12. Simplified 92.6%

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

13. Taylor expanded in v around inf 57.6%

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

    1. *-commutative 57.6%

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

15. Simplified 57.6%

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

16. Final simplification 57.6%

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

Alternative 8: 59.3% accurate, 1.9× speedup

Math

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

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_i (/ v (- sinTheta_O))))
   (* 2.0 (/ v (* cosTheta_i_m 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_i / (v / -sinTheta_O))) / (2.0f * (v / (cosTheta_i_m * 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_i / (v / -sintheta_o))) / (2.0e0 * (v / (costheta_i_m * 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(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(2.0) * Float32(v / Float32(cosTheta_i_m * 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_i / (v / -sinTheta_O))) / (single(2.0) * (v / (cosTheta_i_m * 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* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.7%

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

3. Simplified 92.7%

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

5. Taylor expanded in v around inf 57.5%

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

6. Final simplification 57.5%

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

Alternative 9: 59.3% accurate, 1.9× speedup

Math

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

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_i (/ v (- sinTheta_O))))
   (/ 2.0 (/ 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 * (expf((sinTheta_i / (v / -sinTheta_O))) / (2.0f / (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 * (exp((sintheta_i / (v / -sintheta_o))) / (2.0e0 / (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(exp(Float32(sinTheta_i / Float32(v / Float32(-sinTheta_O)))) / Float32(Float32(2.0) / 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 * (exp((sinTheta_i / (v / -sinTheta_O))) / (single(2.0) / (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. associate-/l* 92.7%

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

   2. *-commutative 92.7%

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

   3. distribute-neg-frac 92.7%

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

   4. distribute-rgt-neg-out 92.7%

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

   5. associate-/l* 92.7%

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

   6. *-commutative 92.7%

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

   7. *-commutative 92.7%

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

   8. associate-/l* 92.7%

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

3. Simplified 92.7%

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

5. Taylor expanded in v around inf 57.5%

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

6. Final simplification 57.5%

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

Alternative 10: 59.2% accurate, 24.4× speedup

Math

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

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 (/ 1.0 (/ v (* cosTheta_i_m 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 * (1.0f / (v / (cosTheta_i_m * 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 * (1.0e0 / (v / (costheta_i_m * 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(Float32(1.0) / Float32(v / Float32(cosTheta_i_m * 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) * (single(1.0) / (v / (cosTheta_i_m * 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 57.1%

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

   1. associate-*l/ 57.1%

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

   2. *-commutative 57.1%

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

7. Simplified 57.1%

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

   1. associate-*r/ 57.1%

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

   2. clear-num 57.4%

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

   3. *-commutative 57.4%

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

9. Applied egg-rr 57.4%

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

10. Final simplification 57.4%

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

Alternative 11: 59.3% accurate, 24.4× speedup

Math

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

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 (/ 1.0 (/ (/ v cosTheta_i_m) (* cosTheta_O 0.5)))))

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 * (1.0f / ((v / cosTheta_i_m) / (cosTheta_O * 0.5f)));
}

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 * (1.0e0 / ((v / costheta_i_m) / (costheta_o * 0.5e0)))
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(1.0) / Float32(Float32(v / cosTheta_i_m) / Float32(cosTheta_O * Float32(0.5)))))
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(1.0) / ((v / cosTheta_i_m) / (cosTheta_O * single(0.5))));
end

Derivation

1. Initial program 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*l* 98.5%

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

   3. times-frac 98.5%

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

   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 57.1%

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

   1. associate-*l/ 57.1%

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

   2. *-commutative 57.1%

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

7. Simplified 57.1%

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

   1. add-exp-log 45.1%

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

   2. clear-num 45.1%

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

   3. div-inv 45.1%

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

   4. associate-/r/ 45.1%

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

9. Applied egg-rr 45.1%

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

10. Taylor expanded in cosTheta_i around 0 17.4%

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

    1. +-commutative 17.4%

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

    2. log-prod 17.4%

       \[\leadsto e^{\color{blue}{\left(\log 0.5 + \log \left(\frac{cosTheta_O}{v}\right)\right)} + \log cosTheta_i} \]

    3. associate-+r+ 17.4%

       \[\leadsto e^{\color{blue}{\log 0.5 + \left(\log \left(\frac{cosTheta_O}{v}\right) + \log cosTheta_i\right)}} \]

    4. +-commutative 17.4%

       \[\leadsto e^{\log 0.5 + \color{blue}{\left(\log cosTheta_i + \log \left(\frac{cosTheta_O}{v}\right)\right)}} \]

    5. log-prod 45.1%

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

    6. associate-*r/ 45.1%

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

    7. associate-*l/ 45.1%

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

    8. log-prod 17.4%

       \[\leadsto e^{\log 0.5 + \color{blue}{\left(\log \left(\frac{cosTheta_i}{v}\right) + \log cosTheta_O\right)}} \]

    9. associate-+l+ 17.4%

       \[\leadsto e^{\color{blue}{\left(\log 0.5 + \log \left(\frac{cosTheta_i}{v}\right)\right) + \log cosTheta_O}} \]

    10. log-prod 17.4%

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

    11. log-prod 45.1%

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

    12. associate-*l* 45.1%

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

    13. associate-*l/ 45.1%

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

    14. *-commutative 45.1%

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

    15. associate-/l* 45.1%

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

    16. associate-*r/ 45.1%

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

12. Simplified 45.1%

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

    1. rem-exp-log 57.1%

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

    2. clear-num 57.5%

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

    3. *-commutative 57.5%

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

14. Applied egg-rr 57.5%

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

15. Final simplification 57.5%

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

Alternative 12: 58.7% accurate, 31.4× speedup

Math

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

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_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(Float32(cosTheta_i_m / 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 57.1%

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

   1. associate-*l/ 57.1%

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

   2. *-commutative 57.1%

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

7. Simplified 57.1%

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

8. Taylor expanded in cosTheta_i around 0 57.1%

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

   1. associate-*r/ 57.1%

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

10. Simplified 57.1%

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

11. Final simplification 57.1%

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

Alternative 13: 59.2% accurate, 31.4× speedup

Math

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

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 (/ 0.5 (/ v (* cosTheta_i_m 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 / (v / (cosTheta_i_m * 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 / (v / (costheta_i_m * 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(v / Float32(cosTheta_i_m * 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) / (v / (cosTheta_i_m * 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 57.1%

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

   1. associate-*l/ 57.1%

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

   2. *-commutative 57.1%

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

7. Simplified 57.1%

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

   1. add-exp-log 45.1%

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

   2. clear-num 45.1%

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

   3. div-inv 45.1%

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

   4. associate-/r/ 45.1%

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

9. Applied egg-rr 45.1%

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

10. Taylor expanded in cosTheta_i around 0 17.4%

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

    1. +-commutative 17.4%

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

    2. log-prod 17.4%

       \[\leadsto e^{\color{blue}{\left(\log 0.5 + \log \left(\frac{cosTheta_O}{v}\right)\right)} + \log cosTheta_i} \]

    3. associate-+r+ 17.4%

       \[\leadsto e^{\color{blue}{\log 0.5 + \left(\log \left(\frac{cosTheta_O}{v}\right) + \log cosTheta_i\right)}} \]

    4. +-commutative 17.4%

       \[\leadsto e^{\log 0.5 + \color{blue}{\left(\log cosTheta_i + \log \left(\frac{cosTheta_O}{v}\right)\right)}} \]

    5. log-prod 45.1%

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

    6. associate-*r/ 45.1%

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

    7. associate-*l/ 45.1%

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

    8. log-prod 17.4%

       \[\leadsto e^{\log 0.5 + \color{blue}{\left(\log \left(\frac{cosTheta_i}{v}\right) + \log cosTheta_O\right)}} \]

    9. associate-+l+ 17.4%

       \[\leadsto e^{\color{blue}{\left(\log 0.5 + \log \left(\frac{cosTheta_i}{v}\right)\right) + \log cosTheta_O}} \]

    10. log-prod 17.4%

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

    11. log-prod 45.1%

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

    12. associate-*l* 45.1%

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

    13. associate-*l/ 45.1%

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

    14. *-commutative 45.1%

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

    15. associate-/l* 45.1%

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

    16. associate-*r/ 45.1%

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

12. Simplified 45.1%

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

    1. rem-exp-log 57.1%

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

    2. associate-/l* 57.4%

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

14. Applied egg-rr 57.4%

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

    1. associate-/l/ 57.4%

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

    2. *-commutative 57.4%

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

16. Simplified 57.4%

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

17. Final simplification 57.4%

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

Reproduce

herbie shell --seed 2024020 
(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)))