HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.8%

Time: 14.4s

Alternatives: 10

Speedup: 1.8×

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, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. associate-/l* 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-/r* 98.5%

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

   2. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. associate-*l/ 98.7%

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

   2. associate-/r* 98.7%

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

   3. +-commutative 98.7%

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

   4. associate-/l* 98.7%

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

10. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.7%

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

    2. frac-2neg 98.7%

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

    3. associate-/l/ 98.7%

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

    4. un-div-inv 98.5%

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

12. Applied egg-rr 98.5%

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

    1. distribute-neg-frac 98.5%

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

    2. neg-mul-1 98.5%

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

    3. associate-*l/ 98.7%

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

    4. *-commutative 98.7%

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

    5. associate-/r* 98.7%

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

    6. metadata-eval 98.7%

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

14. Simplified 98.7%

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

15. Final simplification 98.7%

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

Alternative 2: 98.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. associate-/l* 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-/r* 98.5%

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

   2. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in sinTheta_O around 0 98.6%

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

   1. associate-*l/ 98.7%

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

   2. associate-/r* 98.7%

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

   3. +-commutative 98.7%

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

   4. associate-/l* 98.7%

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

10. Applied egg-rr 98.7%

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

    1. expm1-log1p-u 98.7%

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

    2. expm1-udef 61.9%

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

    3. clear-num 61.9%

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

    4. un-div-inv 61.9%

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

    5. div-inv 61.9%

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

    6. clear-num 61.9%

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

    7. div-inv 61.9%

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

    8. clear-num 61.9%

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

    9. /-rgt-identity 61.9%

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

12. Applied egg-rr 61.9%

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

    1. expm1-def 98.5%

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

    2. expm1-log1p 98.5%

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

    3. associate-/l/ 98.5%

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

    4. *-commutative 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 3: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. associate-/l* 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-/r* 98.5%

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

   2. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in sinTheta_O around 0 98.5%

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

   1. clear-num 95.0%

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

   2. associate-/r* 95.0%

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

   3. frac-2neg 95.0%

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

   4. frac-times 92.6%

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

   5. *-un-lft-identity 92.6%

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

   6. associate-/l/ 92.6%

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

   7. distribute-neg-frac 92.6%

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

   8. metadata-eval 92.6%

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

10. Applied egg-rr 92.6%

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

    1. *-commutative 92.6%

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

    2. associate-*r/ 92.6%

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

    3. associate-/l* 98.4%

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

    4. times-frac 98.5%

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

    5. associate-/r* 98.5%

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

    6. metadata-eval 98.5%

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

12. Simplified 98.5%

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

13. Final simplification 98.5%

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

Alternative 4: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.3%

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

   10. associate-/l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_i around 0 98.3%

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

   1. rec-exp 98.3%

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

   2. distribute-neg-frac 98.3%

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

   3. metadata-eval 98.3%

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

6. Simplified 98.3%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-udef 94.1%

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

   3. div-inv 94.1%

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

   4. add-log-exp 94.1%

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

   5. log-pow 94.1%

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

   6. add-exp-log 94.1%

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

   7. inv-pow 94.1%

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

   8. rec-exp 94.1%

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

   9. sinh-undef 94.1%

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

8. Applied egg-rr 94.1%

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

   1. expm1-def 98.3%

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

   2. expm1-log1p 98.3%

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

   3. associate-/r* 98.3%

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

   4. metadata-eval 98.3%

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

10. Simplified 98.3%

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

11. Final simplification 98.3%

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

Alternative 5: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.3%

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

   10. associate-/l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_i around 0 98.6%

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

5. Taylor expanded in v around inf 64.2%

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

   1. *-commutative 64.2%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{v \cdot 2} + 0.3333333333333333 \cdot \frac{1}{v}} \]

   2. fma-def 64.2%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{\mathsf{fma}\left(v, 2, 0.3333333333333333 \cdot \frac{1}{v}\right)}} \]

   3. associate-*r/ 64.2%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\mathsf{fma}\left(v, 2, \color{blue}{\frac{0.3333333333333333 \cdot 1}{v}}\right)} \]

   4. metadata-eval 64.2%

      \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\mathsf{fma}\left(v, 2, \frac{\color{blue}{0.3333333333333333}}{v}\right)} \]

7. Simplified 64.2%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)}} \]

8. Final simplification 64.2%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{\mathsf{fma}\left(v, 2, \frac{0.3333333333333333}{v}\right)} \]

Alternative 6: 64.7% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.3%

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

   10. associate-/l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in sinTheta_i around 0 98.6%

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

5. Taylor expanded in v around inf 64.2%

   \[\leadsto \frac{cosTheta_O \cdot cosTheta_i}{\color{blue}{2 \cdot v + 0.3333333333333333 \cdot \frac{1}{v}}} \]

6. Final simplification 64.2%

   \[\leadsto \frac{cosTheta_i \cdot cosTheta_O}{v \cdot 2 + \frac{1}{v} \cdot 0.3333333333333333} \]

Alternative 7: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.3%

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

   10. associate-/l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.1%

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

   1. *-commutative 58.1%

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

   2. associate-*l/ 58.1%

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

   3. *-commutative 58.1%

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

6. Simplified 58.1%

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

7. Final simplification 58.1%

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

Alternative 8: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.3%

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

   10. associate-/l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.1%

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

5. Final simplification 58.1%

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

Alternative 9: 58.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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. Step-by-step derivation

   1. div-inv 98.5%

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

   2. associate-/l* 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-/r* 98.5%

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

   2. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in sinTheta_O around 0 98.5%

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

9. Taylor expanded in v around inf 58.1%

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

    1. associate-*r/ 58.1%

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

    2. *-commutative 58.1%

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

    3. associate-*l* 58.1%

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

    4. associate-*l/ 58.1%

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

    5. associate-*r/ 58.1%

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

11. Simplified 58.1%

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

12. Final simplification 58.1%

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

Alternative 10: 59.4% accurate, 31.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. times-frac 98.5%

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

   2. exp-neg 98.5%

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

   3. *-commutative 98.5%

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

   4. exp-neg 98.5%

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

   5. *-commutative 98.5%

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

   6. associate-/l* 98.5%

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

   7. distribute-frac-neg 98.5%

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

   8. associate-/r/ 98.5%

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

   9. associate-/l* 98.3%

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

   10. associate-/l/ 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in v around inf 58.1%

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

   1. clear-num 58.8%

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

   2. un-div-inv 58.8%

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

   3. *-commutative 58.8%

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

6. Applied egg-rr 58.8%

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

7. Final simplification 58.8%

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

Reproduce

herbie shell --seed 2023334 
(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)))