HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 21.9s

Alternatives: 22

Speedup: 1.6×

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: 98.9% accurate, 1.0× speedup

Math

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

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 (/ (exp (- (/ (* sinTheta_i sinTheta_O) v))) v)))
    (/ (sinh (/ 1.0 v)) (/ 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 * (expf(-((sinTheta_i * sinTheta_O) / v)) / v))) / (sinhf((1.0f / v)) / (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 * (exp(-((sintheta_i * sintheta_o) / v)) / v))) / (sinh((1.0e0 / v)) / (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(cosTheta_O_m * Float32(cosTheta_i_m * Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) / v))) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(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 * (exp(-((sinTheta_i * sinTheta_O) / v)) / v))) / (sinh((single(1.0) / v)) / (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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.9

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lower-/.f32 N/A

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

   20. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 99.0

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

   \[\leadsto \frac{cosTheta\_O \cdot \left(cosTheta\_i \cdot \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{v}\right)}{\frac{\sinh \left(\frac{1}{v}\right)}{\frac{0.5}{v}}} \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

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 (/ (exp (- (/ (* sinTheta_i sinTheta_O) v))) v)))
    (* v (* (sinh (/ 1.0 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 * ((cosTheta_O_m * (cosTheta_i_m * (expf(-((sinTheta_i * sinTheta_O) / v)) / v))) / (v * (sinhf((1.0f / 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 * ((costheta_o_m * (costheta_i_m * (exp(-((sintheta_i * sintheta_o) / v)) / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_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(cosTheta_O_m * Float32(cosTheta_i_m * Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) / v))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / 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 * ((cosTheta_O_m * (cosTheta_i_m * (exp(-((sinTheta_i * sinTheta_O) / v)) / v))) / (v * (sinh((single(1.0) / 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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.9

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lower-/.f32 N/A

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

   20. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

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)
    (*
     (/ (exp (- (/ (* sinTheta_i sinTheta_O) v))) v)
     (/ 0.5 (* v (sinh (/ 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) * ((expf(-((sinTheta_i * sinTheta_O) / v)) / v) * (0.5f / (v * sinhf((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) * ((exp(-((sintheta_i * sintheta_o) / v)) / v) * (0.5e0 / (v * sinh((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(cosTheta_O_m * cosTheta_i_m) * Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) / v) * Float32(Float32(0.5) / Float32(v * sinh(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) * ((exp(-((sinTheta_i * sinTheta_O) / v)) / v) * (single(0.5) / (v * sinh((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

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

   2. div-inv N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \left(\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}}{v} \cdot \frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)}\right) \]
6. Add Preprocessing

Alternative 4: 98.6% accurate, 1.6× speedup

Math

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

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 (/ 1.0 v)))
    (/ (sinh (/ 1.0 v)) (/ 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 * (1.0f / v))) / (sinhf((1.0f / v)) / (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 * (1.0e0 / v))) / (sinh((1.0e0 / v)) / (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(cosTheta_O_m * Float32(cosTheta_i_m * Float32(Float32(1.0) / v))) / Float32(sinh(Float32(Float32(1.0) / v)) / Float32(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(1.0) / v))) / (sinh((single(1.0) / v)) / (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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.9

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lower-/.f32 N/A

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

   20. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 99.0

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in sinTheta_i around 0

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

9. Applied rewrites 98.8%

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

Alternative 5: 98.5% accurate, 1.7× speedup

Math

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

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 (fma v v (* sinTheta_O (* sinTheta_i v)))))
    (/ 0.5 (sinh (/ 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 / fmaf(v, v, (sinTheta_O * (sinTheta_i * v))))) * (0.5f / sinhf((1.0f / v)))));
}

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(cosTheta_O_m * Float32(cosTheta_i_m / fma(v, v, Float32(sinTheta_O * Float32(sinTheta_i * v))))) * Float32(Float32(0.5) / sinh(Float32(Float32(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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.9

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lower-/.f32 N/A

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

   20. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. div-inv N/A

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

   7. lower-*.f32 N/A

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 98.7

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

    \[\leadsto \left(cosTheta\_O \cdot \frac{cosTheta\_i}{\mathsf{fma}\left(v, v, sinTheta\_O \cdot \left(sinTheta\_i \cdot v\right)\right)}\right) \cdot \frac{0.5}{\sinh \left(\frac{1}{v}\right)} \]
11. Add Preprocessing

Alternative 6: 98.5% accurate, 1.8× speedup

Math

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

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 (/ 1.0 v)))
    (* v (* (sinh (/ 1.0 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 * ((cosTheta_O_m * (cosTheta_i_m * (1.0f / v))) / (v * (sinhf((1.0f / 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 * ((costheta_o_m * (costheta_i_m * (1.0e0 / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_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(cosTheta_O_m * Float32(cosTheta_i_m * Float32(Float32(1.0) / v))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / 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 * ((cosTheta_O_m * (cosTheta_i_m * (single(1.0) / v))) / (v * (sinh((single(1.0) / 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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.9

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lower-/.f32 N/A

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

   20. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.7

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 7: 98.3% accurate, 1.8× speedup

Math

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

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 (sinh (/ 1.0 v))) (* cosTheta_O_m (/ cosTheta_i_m (* 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 * ((0.5f / sinhf((1.0f / v))) * (cosTheta_O_m * (cosTheta_i_m / (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 * ((0.5e0 / sinh((1.0e0 / v))) * (costheta_o_m * (costheta_i_m / (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(Float32(0.5) / sinh(Float32(Float32(1.0) / v))) * Float32(cosTheta_O_m * Float32(cosTheta_i_m / Float32(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 * ((single(0.5) / sinh((single(1.0) / v))) * (cosTheta_O_m * (cosTheta_i_m / (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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-/.f32 N/A

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

   14. div-inv N/A

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

   15. lower-/.f32 98.9

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

   18. distribute-neg-frac2 N/A

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

   19. lower-/.f32 N/A

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

   20. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. div-inv N/A

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

   7. lower-*.f32 N/A

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. lower-*.f32 98.5

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Alternative 8: 98.4% accurate, 1.8× speedup

Math

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

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 1.0)
    (/ cosTheta_i_m (* v (* (sinh (/ 1.0 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 * ((cosTheta_O_m * 1.0f) * (cosTheta_i_m / (v * (sinhf((1.0f / 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 * ((costheta_o_m * 1.0e0) * (costheta_i_m / (v * (sinh((1.0e0 / 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(cosTheta_O_m * Float32(1.0)) * Float32(cosTheta_i_m / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(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 * ((cosTheta_O_m * single(1.0)) * (cosTheta_i_m / (v * (sinh((single(1.0) / 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

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 9: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i\_s \cdot \left(cosTheta\_O\_s \cdot \frac{cosTheta\_O\_m \cdot cosTheta\_i\_m}{v \cdot \left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, \left(sinTheta\_i \cdot sinTheta\_i\right) \cdot -0.5, -0.16666666666666666\right), \frac{-0.3333333333333333 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, \left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right)\right)\right)}{v}\right)}{-v}\right)}{v} - -2\right)}\right) \end{array} \]

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)
    (*
     v
     (-
      (/
       (fma
        sinTheta_O
        (* sinTheta_i 2.0)
        (/
         (fma
          2.0
          (fma
           (* sinTheta_O sinTheta_O)
           (* (* sinTheta_i sinTheta_i) -0.5)
           -0.16666666666666666)
          (/
           (*
            -0.3333333333333333
            (fma
             sinTheta_O
             sinTheta_i
             (*
              (* sinTheta_O sinTheta_O)
              (* sinTheta_O (* sinTheta_i (* sinTheta_i sinTheta_i))))))
           v))
         (- 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 * ((cosTheta_O_m * cosTheta_i_m) / (v * ((fmaf(sinTheta_O, (sinTheta_i * 2.0f), (fmaf(2.0f, fmaf((sinTheta_O * sinTheta_O), ((sinTheta_i * sinTheta_i) * -0.5f), -0.16666666666666666f), ((-0.3333333333333333f * fmaf(sinTheta_O, sinTheta_i, ((sinTheta_O * sinTheta_O) * (sinTheta_O * (sinTheta_i * (sinTheta_i * sinTheta_i)))))) / v)) / -v)) / v) - -2.0f))));
}

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(cosTheta_O_m * cosTheta_i_m) / Float32(v * Float32(Float32(fma(sinTheta_O, Float32(sinTheta_i * Float32(2.0)), Float32(fma(Float32(2.0), fma(Float32(sinTheta_O * sinTheta_O), Float32(Float32(sinTheta_i * sinTheta_i) * Float32(-0.5)), Float32(-0.16666666666666666)), Float32(Float32(Float32(-0.3333333333333333) * fma(sinTheta_O, sinTheta_i, Float32(Float32(sinTheta_O * sinTheta_O) * Float32(sinTheta_O * Float32(sinTheta_i * Float32(sinTheta_i * sinTheta_i)))))) / v)) / Float32(-v))) / v) - Float32(-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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. associate-/l/ N/A

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

   11. frac-times N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

7. Applied rewrites 67.5%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, \left(sinTheta\_i \cdot sinTheta\_i\right) \cdot -0.5, -0.16666666666666666\right), \frac{-0.3333333333333333 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, \left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right)\right)\right)}{v}\right)}{-v}\right)}{-v} + -2\right) \cdot \left(-v\right)}} \]

8. Final simplification 67.5%

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, \left(sinTheta\_i \cdot sinTheta\_i\right) \cdot -0.5, -0.16666666666666666\right), \frac{-0.3333333333333333 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, \left(sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(sinTheta\_O \cdot \left(sinTheta\_i \cdot \left(sinTheta\_i \cdot sinTheta\_i\right)\right)\right)\right)}{v}\right)}{-v}\right)}{v} - -2\right)} \]
9. Add Preprocessing

Alternative 10: 64.2% accurate, 3.6× speedup

Math

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

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)
    (*
     v
     (*
      (fma
       v
       (* 2.0 (fma sinTheta_i sinTheta_O v))
       (fma
        sinTheta_i
        (* sinTheta_O (* sinTheta_i sinTheta_O))
        0.3333333333333333))
      (/ 1.0 (* 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 * ((cosTheta_O_m * cosTheta_i_m) / (v * (fmaf(v, (2.0f * fmaf(sinTheta_i, sinTheta_O, v)), fmaf(sinTheta_i, (sinTheta_O * (sinTheta_i * sinTheta_O)), 0.3333333333333333f)) * (1.0f / (v * v))))));
}

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(cosTheta_O_m * cosTheta_i_m) / Float32(v * Float32(fma(v, Float32(Float32(2.0) * fma(sinTheta_i, sinTheta_O, v)), fma(sinTheta_i, Float32(sinTheta_O * Float32(sinTheta_i * sinTheta_O)), Float32(0.3333333333333333))) * Float32(Float32(1.0) / Float32(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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. associate-/l/ N/A

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

   11. frac-times N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 67.5%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(-\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), -2, 0.3333333333333333\right)}{v}\right)}{-v} + -2\right)\right)}} \]

8. Taylor expanded in v around 0

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

10. Applied rewrites 67.5%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \frac{\mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_i, sinTheta\_O \cdot sinTheta\_i, 0.3333333333333333\right)\right)}{\color{blue}{v \cdot v}}} \]
11. Step-by-step derivation

12. Applied rewrites 67.5%

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

13. Final simplification 67.5%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(\mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_i, sinTheta\_O, v\right), \mathsf{fma}\left(sinTheta\_i, sinTheta\_O \cdot \left(sinTheta\_i \cdot sinTheta\_O\right), 0.3333333333333333\right)\right) \cdot \frac{1}{v \cdot v}\right)} \]
14. Add Preprocessing

Alternative 11: 64.2% accurate, 3.8× speedup

Math

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

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
   (/
    (* cosTheta_O_m cosTheta_i_m)
    (*
     v
     (/
      (fma
       v
       (* 2.0 (fma sinTheta_O sinTheta_i v))
       (fma
        (* sinTheta_i sinTheta_O)
        (* sinTheta_i sinTheta_O)
        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 * ((cosTheta_O_m * cosTheta_i_m) / (v * (fmaf(v, (2.0f * fmaf(sinTheta_O, sinTheta_i, v)), fmaf((sinTheta_i * sinTheta_O), (sinTheta_i * sinTheta_O), 0.3333333333333333f)) / (v * v)))));
}

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(cosTheta_O_m * cosTheta_i_m) / Float32(v * Float32(fma(v, Float32(Float32(2.0) * fma(sinTheta_O, sinTheta_i, v)), fma(Float32(sinTheta_i * sinTheta_O), Float32(sinTheta_i * sinTheta_O), Float32(0.3333333333333333))) / Float32(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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. associate-/l/ N/A

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

   11. frac-times N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 67.5%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(-\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), -2, 0.3333333333333333\right)}{v}\right)}{-v} + -2\right)\right)}} \]

8. Taylor expanded in v around 0

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

10. Applied rewrites 67.5%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \frac{\mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_i, sinTheta\_O \cdot sinTheta\_i, 0.3333333333333333\right)\right)}{\color{blue}{v \cdot v}}} \]

11. Final simplification 67.5%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \frac{\mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(sinTheta\_i \cdot sinTheta\_O, sinTheta\_i \cdot sinTheta\_O, 0.3333333333333333\right)\right)}{v \cdot v}} \]
12. Add Preprocessing

Alternative 12: 64.2% accurate, 4.5× speedup

Math

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

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)
    (/
     (fma
      v
      (* 2.0 (fma sinTheta_O sinTheta_i v))
      (fma
       (* sinTheta_i sinTheta_O)
       (* sinTheta_i sinTheta_O)
       0.3333333333333333))
     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) / (fmaf(v, (2.0f * fmaf(sinTheta_O, sinTheta_i, v)), fmaf((sinTheta_i * sinTheta_O), (sinTheta_i * sinTheta_O), 0.3333333333333333f)) / v)));
}

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(cosTheta_O_m * cosTheta_i_m) / Float32(fma(v, Float32(Float32(2.0) * fma(sinTheta_O, sinTheta_i, v)), fma(Float32(sinTheta_i * sinTheta_O), Float32(sinTheta_i * sinTheta_O), Float32(0.3333333333333333))) / 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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. associate-/l/ N/A

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

   11. frac-times N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 67.5%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(-\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), -2, 0.3333333333333333\right)}{v}\right)}{-v} + -2\right)\right)}} \]

8. Taylor expanded in v around 0

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

10. Applied rewrites 67.5%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\frac{\mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_i, sinTheta\_O \cdot sinTheta\_i, 0.3333333333333333\right)\right)}{\color{blue}{v}}} \]

11. Final simplification 67.5%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\frac{\mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(sinTheta\_i \cdot sinTheta\_O, sinTheta\_i \cdot sinTheta\_O, 0.3333333333333333\right)\right)}{v}} \]
12. Add Preprocessing

Alternative 13: 64.2% accurate, 5.4× speedup

Math

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

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)
    (fma
     2.0
     (fma sinTheta_O sinTheta_i v)
     (* v (/ 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 * ((cosTheta_O_m * cosTheta_i_m) / fmaf(2.0f, fmaf(sinTheta_O, sinTheta_i, v), (v * (0.3333333333333333f / (v * v))))));
}

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(cosTheta_O_m * cosTheta_i_m) / fma(Float32(2.0), fma(sinTheta_O, sinTheta_i, v), Float32(v * Float32(Float32(0.3333333333333333) / Float32(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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. associate-/l/ N/A

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

   11. frac-times N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 67.5%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(-\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), -2, 0.3333333333333333\right)}{v}\right)}{-v} + -2\right)\right)}} \]

8. Taylor expanded in sinTheta_O around 0

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

10. Applied rewrites 67.5%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\mathsf{fma}\left(2, \color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right)}, \frac{0.3333333333333333}{v \cdot v} \cdot v\right)} \]

11. Final simplification 67.5%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\mathsf{fma}\left(2, \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), v \cdot \frac{0.3333333333333333}{v \cdot v}\right)} \]
12. Add Preprocessing

Alternative 14: 64.2% accurate, 5.6× speedup

Math

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

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
   (/
    (* cosTheta_O_m cosTheta_i_m)
    (* v (/ (fma 2.0 (* v v) 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 * ((cosTheta_O_m * cosTheta_i_m) / (v * (fmaf(2.0f, (v * v), 0.3333333333333333f) / (v * v)))));
}

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(cosTheta_O_m * cosTheta_i_m) / Float32(v * Float32(fma(Float32(2.0), Float32(v * v), Float32(0.3333333333333333)) / Float32(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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. associate-/l/ N/A

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

   11. frac-times N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 67.5%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(-\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), -2, 0.3333333333333333\right)}{v}\right)}{-v} + -2\right)\right)}} \]

8. Taylor expanded in v around 0

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

10. Applied rewrites 67.5%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \frac{\mathsf{fma}\left(v, 2 \cdot \mathsf{fma}\left(sinTheta\_O, sinTheta\_i, v\right), \mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_i, sinTheta\_O \cdot sinTheta\_i, 0.3333333333333333\right)\right)}{\color{blue}{v \cdot v}}} \]

11. Taylor expanded in sinTheta_O around 0

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

13. Applied rewrites 67.5%

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

14. Final simplification 67.5%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \frac{\mathsf{fma}\left(2, v \cdot v, 0.3333333333333333\right)}{v \cdot v}} \]
15. Add Preprocessing

Alternative 15: 64.2% accurate, 6.6× speedup

Math

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

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)
    (* 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 * ((cosTheta_O_m * cosTheta_i_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 * ((costheta_o_m * costheta_i_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(cosTheta_O_m * cosTheta_i_m) / Float32(v * Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(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 * ((cosTheta_O_m * cosTheta_i_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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. exp-neg N/A

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

   10. associate-/l/ N/A

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

   11. frac-times N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 67.5%

   \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(-\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i \cdot 2, \frac{\mathsf{fma}\left(-0.5 \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right), -2, 0.3333333333333333\right)}{v}\right)}{-v} + -2\right)\right)}} \]

8. Taylor expanded in sinTheta_O around 0

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

10. Applied rewrites 67.5%

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

11. Final simplification 67.5%

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

Alternative 16: 58.9% accurate, 8.2× speedup

Math

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

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 (/ 1.0 (/ v (* 0.5 (* 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 * (1.0f / (v / (0.5f * (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 * (1.0e0 / (v / (0.5e0 * (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(1.0) / Float32(v / Float32(Float32(0.5) * 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(1.0) / (v / (single(0.5) * (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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 61.9

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

5. Applied rewrites 61.9%

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

7. Applied rewrites 61.8%

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

9. Applied rewrites 62.4%

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

10. Final simplification 62.4%

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

Alternative 17: 58.9% accurate, 8.2× speedup

Math

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

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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 61.9

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

5. Applied rewrites 61.9%

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

7. Applied rewrites 62.4%

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

Alternative 18: 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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 61.9

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

5. Applied rewrites 61.9%

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

7. Applied rewrites 62.4%

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

Alternative 19: 58.4% accurate, 12.4× speedup

Math

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

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 (/ (* 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) * 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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 61.9

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

5. Applied rewrites 61.9%

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

6. Final simplification 61.9%

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

Alternative 20: 58.4% accurate, 12.4× speedup

Math

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

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_i_m (/ (* cosTheta_O_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_i_m * ((cosTheta_O_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_i_m * ((costheta_o_m * 0.5e0) / v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i\_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(cosTheta_i_m * Float32(Float32(cosTheta_O_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_i_m * ((cosTheta_O_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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 61.9

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

5. Applied rewrites 61.9%

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

7. Applied rewrites 61.8%

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

9. Applied rewrites 61.8%

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

10. Final simplification 61.8%

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

Alternative 21: 58.4% accurate, 12.4× speedup

Math

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

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(0.5) * Float32(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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 61.9

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

5. Applied rewrites 61.9%

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

7. Applied rewrites 61.8%

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

8. Final simplification 61.8%

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

Alternative 22: 58.4% accurate, 12.4× speedup

Math

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

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(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

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 61.9

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

5. Applied rewrites 61.9%

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

7. Applied rewrites 61.8%

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

9. Applied rewrites 61.8%

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

10. Final simplification 61.8%

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

Reproduce

herbie shell --seed 2024226 
(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)))