HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 15.8s

Alternatives: 22

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.9× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* cosTheta_i_m (* cosTheta_O_m (/ -1.0 v))))
    (* (* (/ 1.0 (/ 1.0 (sinh (/ 1.0 v)))) (- 2.0)) v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * (costheta_i_m * (costheta_o_m * ((-1.0e0) / v)))) / (((1.0e0 / (1.0e0 / sinh((1.0e0 / v)))) * -2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(cosTheta_i_m * Float32(cosTheta_O_m * Float32(Float32(-1.0) / v)))) / Float32(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) / sinh(Float32(Float32(1.0) / v)))) * Float32(-Float32(2.0))) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (single(-1.0) / v)))) / (((single(1.0) / (single(1.0) / sinh((single(1.0) / v)))) * -single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f32 N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 N/A

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

   18. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-sinh.f32 N/A

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

   2. sinh-def N/A

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

   3. sinh-undef N/A

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

   4. lift-sinh.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 98.9

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. lift-/.f32 N/A

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

   18. sinh-neg N/A

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

   19. lift-sinh.f32 N/A

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

   20. associate-/r/ N/A

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

   21. metadata-eval N/A

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

   22. frac-2neg N/A

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

   23. associate-/r/ N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

   26. lower-*.f32 N/A

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

8. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lift-sinh.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. div-inv N/A

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

   9. lift-/.f32 N/A

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

   10. neg-mul-1 N/A

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

   11. sinh-neg N/A

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

   12. lift-sinh.f32 N/A

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

   13. frac-2neg N/A

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

   14. lower-/.f32 98.9

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* cosTheta_i_m (* cosTheta_O_m (/ -1.0 v))))
    (* (* (sinh (/ 1.0 v)) (- 2.0)) v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * (costheta_i_m * (costheta_o_m * ((-1.0e0) / v)))) / ((sinh((1.0e0 / v)) * -2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(cosTheta_i_m * Float32(cosTheta_O_m * Float32(Float32(-1.0) / v)))) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(-Float32(2.0))) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (single(-1.0) / v)))) / ((sinh((single(1.0) / v)) * -single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f32 N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 N/A

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

   18. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* cosTheta_O_m (* cosTheta_i_m (/ -1.0 v))))
    (* (* (sinh (/ 1.0 v)) (- 2.0)) v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * (costheta_o_m * (costheta_i_m * ((-1.0e0) / v)))) / ((sinh((1.0e0 / v)) * -2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(cosTheta_O_m * Float32(cosTheta_i_m * Float32(Float32(-1.0) / v)))) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(-Float32(2.0))) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_O_m * (cosTheta_i_m * (single(-1.0) / v)))) / ((sinh((single(1.0) / v)) * -single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

   15. lift-/.f32 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f32 N/A

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

   18. lower-neg.f32 N/A

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

   19. lower-*.f32 98.8

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* (/ 1.0 v) (* cosTheta_O_m cosTheta_i_m)))
    (* (* (sinh (/ 1.0 v)) 2.0) v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * ((1.0e0 / v) * (costheta_o_m * costheta_i_m))) / ((sinh((1.0e0 / v)) * 2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_O_m * cosTheta_i_m))) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((single(1.0) / v) * (cosTheta_O_m * cosTheta_i_m))) / ((sinh((single(1.0) / v)) * single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* (/ cosTheta_O_m v) cosTheta_i_m))
    (* (* (sinh (/ 1.0 v)) 2.0) v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * ((costheta_o_m / v) * costheta_i_m)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(cosTheta_O_m / v) * cosTheta_i_m)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((cosTheta_O_m / v) * cosTheta_i_m)) / ((sinh((single(1.0) / v)) * single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (* (/ (/ cosTheta_i_m v) v) cosTheta_O_m)
    (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((((costheta_i_m / v) / v) * costheta_o_m) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(Float32(cosTheta_i_m / v) / v) * cosTheta_O_m) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((((cosTheta_i_m / v) / v) * cosTheta_O_m) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. rec-exp N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower--.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

   18. lower-exp.f32 N/A

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

   19. distribute-neg-frac N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f32 98.6

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

8. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
9. Add Preprocessing

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (* (/ (/ cosTheta_O_m v) v) cosTheta_i_m)
    (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((((costheta_o_m / v) / v) * costheta_i_m) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(Float32(cosTheta_O_m / v) / v) * cosTheta_i_m) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((((cosTheta_O_m / v) / v) * cosTheta_i_m) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)))));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

9. Taylor expanded in sinTheta_i around 0

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

    1. times-frac N/A

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

    2. associate-*r/ N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. unpow2 N/A

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

    6. associate-/r* N/A

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

    7. lower-/.f32 N/A

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

    8. lower-/.f32 N/A

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

    9. rec-exp N/A

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

    10. distribute-neg-frac N/A

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

    11. metadata-eval N/A

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

    12. lower--.f32 N/A

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

    13. lower-exp.f32 N/A

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

    14. lower-/.f32 N/A

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

    15. metadata-eval N/A

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

    16. distribute-neg-frac N/A

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

    17. lower-exp.f32 N/A

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

    18. distribute-neg-frac N/A

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

    19. metadata-eval N/A

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

    20. lower-/.f32 98.7

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

11. Applied rewrites 98.7%

    \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_O}{v}}{v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
12. Add Preprocessing

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (/ (* (- cosTheta_O_m) cosTheta_i_m) (* v v))
    (- (exp (/ -1.0 v)) (exp (/ 1.0 v)))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * (((-costheta_o_m * costheta_i_m) / (v * v)) / (exp(((-1.0e0) / v)) - exp((1.0e0 / v)))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(Float32(-cosTheta_O_m) * cosTheta_i_m) / Float32(v * v)) / Float32(exp(Float32(Float32(-1.0) / v)) - exp(Float32(Float32(1.0) / v))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((-cosTheta_O_m * cosTheta_i_m) / (v * v)) / (exp((single(-1.0) / v)) - exp((single(1.0) / v)))));
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. sinh-neg N/A

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

   12. lower-sinh.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f32 N/A

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

   15. metadata-eval 98.7

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. rec-exp N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. lower--.f32 N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. lower-exp.f32 N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 N/A

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

   21. lower-exp.f32 N/A

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

   22. lower-/.f32 98.6

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

9. Applied rewrites 98.6%

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

Alternative 9: 70.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(cosTheta\_i\_m \cdot \left(cosTheta\_O\_m \cdot \frac{-1}{v}\right)\right)}{\left(\frac{-1 - \frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v}}{v}}{-v} \cdot \left(-2\right)\right) \cdot v}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* cosTheta_i_m (* cosTheta_O_m (/ -1.0 v))))
    (*
     (*
      (/
       (-
        -1.0
        (/ (/ (+ (/ 0.008333333333333333 (* v v)) 0.16666666666666666) v) v))
       (- v))
      (- 2.0))
     v)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * ((expf(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (-1.0f / v)))) / ((((-1.0f - ((((0.008333333333333333f / (v * v)) + 0.16666666666666666f) / v) / v)) / -v) * -2.0f) * v)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * (costheta_i_m * (costheta_o_m * ((-1.0e0) / v)))) / (((((-1.0e0) - ((((0.008333333333333333e0 / (v * v)) + 0.16666666666666666e0) / v) / v)) / -v) * -2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(cosTheta_i_m * Float32(cosTheta_O_m * Float32(Float32(-1.0) / v)))) / Float32(Float32(Float32(Float32(Float32(-1.0) - Float32(Float32(Float32(Float32(Float32(0.008333333333333333) / Float32(v * v)) + Float32(0.16666666666666666)) / v) / v)) / Float32(-v)) * Float32(-Float32(2.0))) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (single(-1.0) / v)))) / ((((single(-1.0) - ((((single(0.008333333333333333) / (v * v)) + single(0.16666666666666666)) / v) / v)) / -v) * -single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f32 N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 N/A

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

   18. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)} \cdot 2\right) \cdot v} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}} \cdot 2\right) \cdot v} \]

   3. neg-mul-1 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\color{blue}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{-1 \cdot v}} \cdot 2\right) \cdot v} \]

9. Applied rewrites 71.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\color{blue}{\frac{-1 - \frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v}}{v}}{-v}} \cdot 2\right) \cdot v} \]

10. Final simplification 71.2%

    \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{-1 - \frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v}}{v}}{-v} \cdot \left(-2\right)\right) \cdot v} \]
11. Add Preprocessing

Alternative 10: 70.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \frac{cosTheta\_i\_m \cdot cosTheta\_O\_m}{v}}{\left(\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v}}{v} - -1}{v} \cdot 2\right) \cdot v}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (/ (* cosTheta_i_m cosTheta_O_m) v))
    (*
     (*
      (/
       (-
        (/ (/ (- (/ 0.008333333333333333 (* v v)) -0.16666666666666666) v) v)
        -1.0)
       v)
      2.0)
     v)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * ((expf(((sinTheta_i * sinTheta_O) / -v)) * ((cosTheta_i_m * cosTheta_O_m) / v)) / ((((((((0.008333333333333333f / (v * v)) - -0.16666666666666666f) / v) / v) - -1.0f) / v) * 2.0f) * v)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * ((costheta_i_m * costheta_o_m) / v)) / ((((((((0.008333333333333333e0 / (v * v)) - (-0.16666666666666666e0)) / v) / v) - (-1.0e0)) / v) * 2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(cosTheta_i_m * cosTheta_O_m) / v)) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.008333333333333333) / Float32(v * v)) - Float32(-0.16666666666666666)) / v) / v) - Float32(-1.0)) / v) * Float32(2.0)) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((cosTheta_i_m * cosTheta_O_m) / v)) / ((((((((single(0.008333333333333333) / (v * v)) - single(-0.16666666666666666)) / v) / v) - single(-1.0)) / v) * single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)}{v}} \cdot 2\right) \cdot v} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1\right)\right)}}{v} \cdot 2\right) \cdot v} \]

   3. sub-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{v} \cdot 2\right) \cdot v} \]

   4. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\left(\color{blue}{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} \cdot -1} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)}{v} \cdot 2\right) \cdot v} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} \cdot -1 + \color{blue}{-1}\right)\right)}{v} \cdot 2\right) \cdot v} \]

   6. distribute-lft1-in N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1\right) \cdot -1}\right)}{v} \cdot 2\right) \cdot v} \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1\right) \cdot \left(\mathsf{neg}\left(-1\right)\right)}}{v} \cdot 2\right) \cdot v} \]

   8. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\left(\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1\right) \cdot \color{blue}{1}}{v} \cdot 2\right) \cdot v} \]

   9. *-rgt-identity N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\color{blue}{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1}}{v} \cdot 2\right) \cdot v} \]

   10. lower-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{\frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + 1}{v}} \cdot 2\right) \cdot v} \]

5. Applied rewrites 71.2%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v}}{v} - -1}{v}} \cdot 2\right) \cdot v} \]

6. Final simplification 71.2%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\frac{\frac{\frac{\frac{0.008333333333333333}{v \cdot v} - -0.16666666666666666}{v}}{v} - -1}{v} \cdot 2\right) \cdot v} \]
7. Add Preprocessing

Alternative 11: 64.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(cosTheta\_i\_m \cdot \left(cosTheta\_O\_m \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v} \cdot \left(-2\right)\right) \cdot v}\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* cosTheta_i_m (* cosTheta_O_m (/ -1.0 v))))
    (* (* (/ (+ (/ 0.16666666666666666 (* v v)) 1.0) v) (- 2.0)) v)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i_s, float cosTheta_O_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i_s * (cosTheta_O_s * ((expf(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (-1.0f / v)))) / (((((0.16666666666666666f / (v * v)) + 1.0f) / v) * -2.0f) * v)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * (costheta_i_m * (costheta_o_m * ((-1.0e0) / v)))) / (((((0.16666666666666666e0 / (v * v)) + 1.0e0) / v) * -2.0e0) * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(cosTheta_i_m * Float32(cosTheta_O_m * Float32(Float32(-1.0) / v)))) / Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(v * v)) + Float32(1.0)) / v) * Float32(-Float32(2.0))) * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (cosTheta_i_m * (cosTheta_O_m * (single(-1.0) / v)))) / (((((single(0.16666666666666666) / (v * v)) + single(1.0)) / v) * -single(2.0)) * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f32 N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 N/A

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

   18. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}} \cdot 2\right) \cdot v} \]

   2. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1}}{v} \cdot 2\right) \cdot v} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1}}{v} \cdot 2\right) \cdot v} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\color{blue}{\frac{\frac{1}{6} \cdot 1}{{v}^{2}}} + 1}{v} \cdot 2\right) \cdot v} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\frac{\color{blue}{\frac{1}{6}}}{{v}^{2}} + 1}{v} \cdot 2\right) \cdot v} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\color{blue}{\frac{\frac{1}{6}}{{v}^{2}}} + 1}{v} \cdot 2\right) \cdot v} \]

   7. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\frac{\frac{1}{6}}{\color{blue}{v \cdot v}} + 1}{v} \cdot 2\right) \cdot v} \]

   8. lower-*.f32 65.4

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\frac{0.16666666666666666}{\color{blue}{v \cdot v}} + 1}{v} \cdot 2\right) \cdot v} \]

9. Applied rewrites 65.4%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\color{blue}{\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v}} \cdot 2\right) \cdot v} \]

10. Final simplification 65.4%

    \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \left(cosTheta\_i \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\left(\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v} \cdot \left(-2\right)\right) \cdot v} \]
11. Add Preprocessing

Alternative 12: 64.0% accurate, 1.5× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (* (- cosTheta_i_m) (* cosTheta_O_m (/ -1.0 v))))
    (+ (/ 0.3333333333333333 (* v v)) 2.0)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * (-costheta_i_m * (costheta_o_m * ((-1.0e0) / v)))) / ((0.3333333333333333e0 / (v * v)) + 2.0e0)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(-cosTheta_i_m) * Float32(cosTheta_O_m * Float32(Float32(-1.0) / v)))) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * (-cosTheta_i_m * (cosTheta_O_m * (single(-1.0) / v)))) / ((single(0.3333333333333333) / (v * v)) + single(2.0))));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.8

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. div-inv N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. frac-2neg N/A

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

   14. lift-/.f32 N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f32 N/A

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

   17. lower-neg.f32 N/A

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

   18. lower-*.f32 98.9

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

6. Applied rewrites 98.9%

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

   1. lift-sinh.f32 N/A

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

   2. sinh-def N/A

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

   3. sinh-undef N/A

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

   4. lift-sinh.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 98.9

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lift-sinh.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. lift-/.f32 N/A

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

   18. sinh-neg N/A

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

   19. lift-sinh.f32 N/A

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

   20. associate-/r/ N/A

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

   21. metadata-eval N/A

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

   22. frac-2neg N/A

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

   23. associate-/r/ N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

   26. lower-*.f32 N/A

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

    2. lower-+.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

    4. metadata-eval N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

    5. lower-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

    6. unpow2 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(-cosTheta\_i\right) \cdot \left(cosTheta\_O \cdot \frac{-1}{v}\right)\right)}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

    7. lower-*.f32 65.4

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

11. Applied rewrites 65.4%

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

12. Final simplification 65.4%

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

Alternative 13: 64.0% accurate, 1.5× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (/ (* cosTheta_i_m cosTheta_O_m) v))
    (- 2.0 (/ (/ -0.3333333333333333 v) v))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * ((costheta_i_m * costheta_o_m) / v)) / (2.0e0 - (((-0.3333333333333333e0) / v) / v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(cosTheta_i_m * cosTheta_O_m) / v)) / Float32(Float32(2.0) - Float32(Float32(Float32(-0.3333333333333333) / v) / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((cosTheta_i_m * cosTheta_O_m) / v)) / (single(2.0) - ((single(-0.3333333333333333) / v) / v))));
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-sinh.f32 N/A

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

   6. sinh-undef N/A

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

   7. sub-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-exp.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. distribute-neg-frac N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f32 98.7

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

4. Applied rewrites 98.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{1 + \left(\log \mathsf{E}\left(\right) + -1 \cdot \frac{\frac{1}{2} + \left(-1 \cdot \frac{\frac{1}{6} + \frac{1}{6} \cdot {\log \mathsf{E}\left(\right)}^{3}}{v} + \frac{-1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)}{v}\right)}} \]

9. Applied rewrites 65.4%

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

10. Final simplification 65.4%

    \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 - \frac{\frac{-0.3333333333333333}{v}}{v}} \]
11. Add Preprocessing

Alternative 14: 64.0% accurate, 1.6× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (*
   cosTheta_O_s
   (/
    (*
     (exp (/ (* sinTheta_i sinTheta_O) (- v)))
     (/ (* cosTheta_i_m cosTheta_O_m) v))
    (+ (/ 0.3333333333333333 (* v v)) 2.0)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((exp(((sintheta_i * sintheta_o) / -v)) * ((costheta_i_m * costheta_o_m) / v)) / ((0.3333333333333333e0 / (v * v)) + 2.0e0)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) * Float32(Float32(cosTheta_i_m * cosTheta_O_m) / v)) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((exp(((sinTheta_i * sinTheta_O) / -v)) * ((cosTheta_i_m * cosTheta_O_m) / v)) / ((single(0.3333333333333333) / (v * v)) + single(2.0))));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 65.4

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

5. Applied rewrites 65.4%

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

6. Final simplification 65.4%

   \[\leadsto \frac{e^{\frac{sinTheta\_i \cdot sinTheta\_O}{-v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
7. Add Preprocessing

Alternative 15: 58.8% accurate, 8.0× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_O_s (/ 0.5 (/ (/ v cosTheta_i_m) cosTheta_O_m)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * (0.5e0 / ((v / costheta_i_m) / costheta_o_m)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(0.5) / Float32(Float32(v / cosTheta_i_m) / cosTheta_O_m))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (single(0.5) / ((v / cosTheta_i_m) / cosTheta_O_m)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

10. Applied rewrites 60.2%

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

12. Applied rewrites 60.6%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{v}{cosTheta\_i}}{cosTheta\_O}}} \]
13. Add Preprocessing

Alternative 16: 58.8% accurate, 9.7× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_O_s (/ 0.5 (/ v (* cosTheta_O_m cosTheta_i_m))))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * (0.5e0 / (v / (costheta_o_m * costheta_i_m))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(0.5) / Float32(v / Float32(cosTheta_O_m * cosTheta_i_m)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (single(0.5) / (v / (cosTheta_O_m * cosTheta_i_m))));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

10. Applied rewrites 60.6%

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

Alternative 17: 58.3% accurate, 10.1× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_i_s
  (* cosTheta_O_s (* (* (* cosTheta_O_m cosTheta_i_m) 0.5) (/ 1.0 v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * (((costheta_o_m * costheta_i_m) * 0.5e0) * (1.0e0 / v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_O_m * cosTheta_i_m) * Float32(0.5)) * Float32(Float32(1.0) / v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((cosTheta_O_m * cosTheta_i_m) * single(0.5)) * (single(1.0) / v)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

10. Applied rewrites 60.2%

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

Alternative 18: 58.3% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_O_s (/ (* (* cosTheta_O_m cosTheta_i_m) 0.5) v))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * (((costheta_o_m * costheta_i_m) * 0.5e0) / v))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_O_m * cosTheta_i_m) * Float32(0.5)) / v)))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((cosTheta_O_m * cosTheta_i_m) * single(0.5)) / v));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

10. Applied rewrites 60.2%

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

Alternative 19: 58.2% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_O_s (* (* (/ cosTheta_i_m v) cosTheta_O_m) 0.5))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * (((costheta_i_m / v) * costheta_o_m) * 0.5e0))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(cosTheta_i_m / v) * cosTheta_O_m) * Float32(0.5))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (((cosTheta_i_m / v) * cosTheta_O_m) * single(0.5)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

10. Applied rewrites 60.2%

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

12. Applied rewrites 60.2%

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

Alternative 20: 58.2% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_O_s (* (* 0.5 cosTheta_i_m) (/ cosTheta_O_m v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((0.5e0 * costheta_i_m) * (costheta_o_m / v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(0.5) * cosTheta_i_m) * Float32(cosTheta_O_m / v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((single(0.5) * cosTheta_i_m) * (cosTheta_O_m / v)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

10. Applied rewrites 60.2%

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

Alternative 21: 58.2% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_O_s (* (* 0.5 cosTheta_O_m) (/ cosTheta_i_m v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * ((0.5e0 * costheta_o_m) * (costheta_i_m / v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(Float32(0.5) * cosTheta_O_m) * Float32(cosTheta_i_m / v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * ((single(0.5) * cosTheta_O_m) * (cosTheta_i_m / v)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

10. Applied rewrites 60.2%

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

Alternative 22: 58.2% accurate, 12.4× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
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_i_s cosTheta_O_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i_s (* cosTheta_O_s (* 0.5 (/ (* cosTheta_O_m cosTheta_i_m) v)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
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_i_s, costheta_o_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_o_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_i_s * (costheta_o_s * (0.5e0 * ((costheta_o_m * costheta_i_m) / v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i_s * Float32(cosTheta_O_s * Float32(Float32(0.5) * Float32(Float32(cosTheta_O_m * cosTheta_i_m) / v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * 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_i_s, cosTheta_O_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i_s * (cosTheta_O_s * (single(0.5) * ((cosTheta_O_m * cosTheta_i_m) / v)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

5. Applied rewrites 60.1%

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

6. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 60.1

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

8. Applied rewrites 60.1%

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

Reproduce

herbie shell --seed 2024314 
(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)))