HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.7%

Time: 26.3s

Alternatives: 27

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\left(\left(\frac{cosTheta\_O\_m}{v} \cdot cosTheta\_i\_m\right) \cdot 0.5\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* (* (/ cosTheta_O_m v) cosTheta_i_m) 0.5)
    (/
     (/ (exp (fma sinTheta_i (- (/ sinTheta_O v)) 0.0)) v)
     (sinh (/ 1.0 v)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((((cosTheta_O_m / v) * cosTheta_i_m) * 0.5f) * ((expf(fmaf(sinTheta_i, -(sinTheta_O / v), 0.0f)) / v) / sinhf((1.0f / v)))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(cosTheta_O_m / v) * cosTheta_i_m) * Float32(0.5)) * Float32(Float32(exp(fma(sinTheta_i, Float32(-Float32(sinTheta_O / v)), Float32(0.0))) / v) / sinh(Float32(Float32(1.0) / v))))))
end

Derivation

1. Initial program 98.3%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. associate-/l/ N/A

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

   8. *-lowering-*.f32 N/A

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

   9. div-inv N/A

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

   10. *-lowering-*.f32 N/A

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

   11. /-lowering-/.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. metadata-eval N/A

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

   14. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot 0.5\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)}} \]
5. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(\color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)} \cdot \frac{1}{2}\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)} \cdot \frac{1}{2}\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)} \cdot \frac{1}{2}\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)} \]

   4. /-lowering-/.f32 98.7

      \[\leadsto \left(\left(\color{blue}{\frac{cosTheta\_O}{v}} \cdot cosTheta\_i\right) \cdot 0.5\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)} \]

6. Applied egg-rr 98.7%

   \[\leadsto \left(\color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)} \cdot 0.5\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)} \]

7. Final simplification 98.7%

   \[\leadsto \left(\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot 0.5\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right)} \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\left(cosTheta\_O\_m \cdot cosTheta\_i\_m\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* cosTheta_O_m cosTheta_i_m)
    (/
     (/ (exp (fma sinTheta_i (- (/ sinTheta_O v)) 0.0)) v)
     (* (sinh (/ 1.0 v)) (* v 2.0)))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * cosTheta_i_m) * ((expf(fmaf(sinTheta_i, -(sinTheta_O / v), 0.0f)) / v) / (sinhf((1.0f / v)) * (v * 2.0f)))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * cosTheta_i_m) * Float32(Float32(exp(fma(sinTheta_i, Float32(-Float32(sinTheta_O / v)), Float32(0.0))) / v) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(2.0)))))))
end

Derivation

1. Initial program 98.3%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/l* N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]

5. Final simplification 98.6%

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)} \]
6. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\left(cosTheta\_O\_m \cdot cosTheta\_i\_m\right) \cdot \left(\frac{e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}}{v} \cdot \frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)}\right)\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* cosTheta_O_m cosTheta_i_m)
    (*
     (/ (exp (fma sinTheta_i (- (/ sinTheta_O v)) 0.0)) v)
     (/ 0.5 (* v (sinh (/ 1.0 v)))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * cosTheta_i_m) * ((expf(fmaf(sinTheta_i, -(sinTheta_O / v), 0.0f)) / v) * (0.5f / (v * sinhf((1.0f / v)))))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * cosTheta_i_m) * Float32(Float32(exp(fma(sinTheta_i, Float32(-Float32(sinTheta_O / v)), Float32(0.0))) / v) * Float32(Float32(0.5) / Float32(v * sinh(Float32(Float32(1.0) / v))))))))
end

Derivation

1. Initial program 98.3%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.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 egg-rr 98.6%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \left(\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v} \cdot \frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)}\right)} \]

5. Final simplification 98.6%

   \[\leadsto \left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \left(\frac{e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}}{v} \cdot \frac{0.5}{v \cdot \sinh \left(\frac{1}{v}\right)}\right) \]
6. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\frac{cosTheta\_O\_m}{v} \cdot \frac{cosTheta\_i\_m}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/ cosTheta_O_m v)
    (/
     cosTheta_i_m
     (*
      (sinh (/ 1.0 v))
      (* (* v 2.0) (exp (fma sinTheta_i (/ sinTheta_O v) 0.0)))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m / v) * (cosTheta_i_m / (sinhf((1.0f / v)) * ((v * 2.0f) * expf(fmaf(sinTheta_i, (sinTheta_O / v), 0.0f)))))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m / v) * Float32(cosTheta_i_m / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(Float32(v * Float32(2.0)) * exp(fma(sinTheta_i, Float32(sinTheta_O / v), Float32(0.0)))))))))
end

Derivation

1. Initial program 98.3%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-lowering-*.f32 N/A

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

   9. /-lowering-/.f32 N/A

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

   10. *-commutative N/A

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

   11. /-rgt-identity N/A

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

   12. associate-/r/ N/A

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

   13. exp-neg N/A

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_O}{v} \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)}} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(cosTheta\_i\_m \cdot \frac{\frac{cosTheta\_O\_m}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    cosTheta_i_m
    (/
     (/ cosTheta_O_m v)
     (*
      (sinh (/ 1.0 v))
      (* (* v 2.0) (exp (fma sinTheta_i (/ sinTheta_O v) 0.0)))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (cosTheta_i_m * ((cosTheta_O_m / v) / (sinhf((1.0f / v)) * ((v * 2.0f) * expf(fmaf(sinTheta_i, (sinTheta_O / v), 0.0f)))))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(cosTheta_i_m * Float32(Float32(cosTheta_O_m / v) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(Float32(v * Float32(2.0)) * exp(fma(sinTheta_i, Float32(sinTheta_O / v), Float32(0.0)))))))))
end

Derivation

1. Initial program 98.3%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. associate-/l* N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-commutative N/A

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

   9. /-rgt-identity N/A

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

   10. associate-/r/ N/A

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

   11. exp-neg N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{cosTheta\_i \cdot \frac{\frac{cosTheta\_O}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)}} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(cosTheta\_O\_m \cdot \frac{\frac{cosTheta\_i\_m}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    cosTheta_O_m
    (/
     (/ cosTheta_i_m v)
     (*
      (sinh (/ 1.0 v))
      (* (* v 2.0) (exp (fma sinTheta_i (/ sinTheta_O v) 0.0)))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (cosTheta_O_m * ((cosTheta_i_m / v) / (sinhf((1.0f / v)) * ((v * 2.0f) * expf(fmaf(sinTheta_i, (sinTheta_O / v), 0.0f)))))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(cosTheta_O_m * Float32(Float32(cosTheta_i_m / v) / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(Float32(v * Float32(2.0)) * exp(fma(sinTheta_i, Float32(sinTheta_O / v), Float32(0.0)))))))))
end

Derivation

1. Initial program 98.3%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. associate-/r* N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. associate-/l* N/A

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

   8. *-lowering-*.f32 N/A

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

   9. *-commutative N/A

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

   10. /-rgt-identity N/A

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

   11. associate-/r/ N/A

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{cosTheta\_O \cdot \frac{\frac{cosTheta\_i}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)}\right)}} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\left(cosTheta\_i\_m \cdot e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}\right) \cdot \frac{cosTheta\_O\_m}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* cosTheta_i_m (exp (fma sinTheta_i (- (/ sinTheta_O v)) 0.0)))
    (/ cosTheta_O_m (* (sinh (/ 1.0 v)) (* v (* v 2.0))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m * expf(fmaf(sinTheta_i, -(sinTheta_O / v), 0.0f))) * (cosTheta_O_m / (sinhf((1.0f / v)) * (v * (v * 2.0f))))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_i_m * exp(fma(sinTheta_i, Float32(-Float32(sinTheta_O / v)), Float32(0.0)))) * Float32(cosTheta_O_m / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0))))))))
end

Derivation

1. Initial program 98.3%

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. exp-lowering-exp.f32 N/A

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

   7. +-lft-identity N/A

      \[\leadsto \left(e^{\color{blue}{0 + \left(\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right)}} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. +-commutative N/A

      \[\leadsto \left(e^{\color{blue}{\left(\mathsf{neg}\left(\frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right) + 0}} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. associate-/l* N/A

      \[\leadsto \left(e^{\left(\mathsf{neg}\left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\right)\right) + 0} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

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

       \[\leadsto \left(e^{\color{blue}{sinTheta\_i \cdot \left(\mathsf{neg}\left(\frac{sinTheta\_O}{v}\right)\right)} + 0} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. accelerator-lowering-fma.f32 N/A

       \[\leadsto \left(e^{\color{blue}{\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(\frac{sinTheta\_O}{v}\right), 0\right)}} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. neg-sub0 N/A

       \[\leadsto \left(e^{\mathsf{fma}\left(sinTheta\_i, \color{blue}{0 - \frac{sinTheta\_O}{v}}, 0\right)} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. --lowering--.f32 N/A

       \[\leadsto \left(e^{\mathsf{fma}\left(sinTheta\_i, \color{blue}{0 - \frac{sinTheta\_O}{v}}, 0\right)} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \left(e^{\mathsf{fma}\left(sinTheta\_i, 0 - \color{blue}{\frac{sinTheta\_O}{v}}, 0\right)} \cdot cosTheta\_i\right) \cdot \frac{\frac{cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. associate-/l/ N/A

       \[\leadsto \left(e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i\right) \cdot \color{blue}{\frac{cosTheta\_O}{\left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v\right) \cdot v}} \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_i\right) \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)}} \]

5. Final simplification 98.5%

   \[\leadsto \left(cosTheta\_i \cdot e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}\right) \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \]
6. Add Preprocessing

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(\left(cosTheta\_O\_m \cdot e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}\right) \cdot \frac{cosTheta\_i\_m}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* cosTheta_O_m (exp (fma sinTheta_i (- (/ sinTheta_O v)) 0.0)))
    (/ cosTheta_i_m (* (sinh (/ 1.0 v)) (* v (* v 2.0))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * expf(fmaf(sinTheta_i, -(sinTheta_O / v), 0.0f))) * (cosTheta_i_m / (sinhf((1.0f / v)) * (v * (v * 2.0f))))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(cosTheta_O_m * exp(fma(sinTheta_i, Float32(-Float32(sinTheta_O / v)), Float32(0.0)))) * Float32(cosTheta_i_m / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(v * Float32(v * Float32(2.0))))))))
end

Derivation

1. Initial program 98.3%

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

   1. associate-*r/ N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.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 egg-rr 98.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)} \cdot cosTheta\_O\right) \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(\left(v \cdot 2\right) \cdot v\right)}} \]

5. Final simplification 98.4%

   \[\leadsto \left(cosTheta\_O \cdot e^{\mathsf{fma}\left(sinTheta\_i, -\frac{sinTheta\_O}{v}, 0\right)}\right) \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot \left(v \cdot 2\right)\right)} \]
6. Add Preprocessing

Alternative 9: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \left(cosTheta\_i\_m \cdot \frac{cosTheta\_O\_m}{\sinh \left(\frac{1}{v}\right) \cdot \left(e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)}\right)\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    cosTheta_i_m
    (/
     cosTheta_O_m
     (*
      (sinh (/ 1.0 v))
      (* (exp (fma sinTheta_i (/ sinTheta_O v) 0.0)) (* v (* v 2.0)))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (cosTheta_i_m * (cosTheta_O_m / (sinhf((1.0f / v)) * (expf(fmaf(sinTheta_i, (sinTheta_O / v), 0.0f)) * (v * (v * 2.0f)))))));
}

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(cosTheta_i_m * Float32(cosTheta_O_m / Float32(sinh(Float32(Float32(1.0) / v)) * Float32(exp(fma(sinTheta_i, Float32(sinTheta_O / v), Float32(0.0))) * Float32(v * Float32(v * Float32(2.0)))))))))
end

Derivation

1. Initial program 98.3%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/l* N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(\left(v \cdot 2\right) \cdot v\right)\right)} \cdot cosTheta\_i} \]

7. Final simplification 98.5%

   \[\leadsto cosTheta\_i \cdot \frac{cosTheta\_O}{\sinh \left(\frac{1}{v}\right) \cdot \left(e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot \left(v \cdot 2\right)\right)\right)} \]
8. Add Preprocessing

Alternative 10: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

   1. times-frac N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. 1-exp N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{e^{0}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{e^{0}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. 1-exp 98.4

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

9. Applied egg-rr 98.4%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-lowering-*.f32 N/A

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

    4. *-lowering-*.f32 N/A

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

    5. /-lowering-/.f32 N/A

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

    6. /-lowering-/.f32 98.4

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

11. Applied egg-rr 98.4%

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

12. Final simplification 98.4%

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

Alternative 11: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/l* N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{e^{\mathsf{fma}\left(sinTheta\_i, 0 - \frac{sinTheta\_O}{v}, 0\right)}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. /-lowering-/.f32 98.4

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Alternative 12: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

   1. times-frac N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. 1-exp N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{e^{0}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{e^{0}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. 1-exp 98.4

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

9. Applied egg-rr 98.4%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. *-lowering-*.f32 N/A

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

    5. *-lowering-*.f32 N/A

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

    6. /-lowering-/.f32 N/A

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

    7. associate-/l/ N/A

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

    8. /-lowering-/.f32 N/A

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

    9. *-lowering-*.f32 N/A

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

    10. *-lowering-*.f32 N/A

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

    11. sinh-lowering-sinh.f32 N/A

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

    12. /-lowering-/.f32 98.3

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

11. Applied egg-rr 98.3%

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

12. Final simplification 98.3%

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

Alternative 13: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

   1. times-frac N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. 1-exp N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{e^{0}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{e^{0}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. 1-exp 98.4

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

9. Applied egg-rr 98.4%

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

    1. times-frac N/A

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

    2. div-inv N/A

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

    3. metadata-eval N/A

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

    4. associate-/r/ N/A

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

    5. clear-num N/A

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

    6. *-lowering-*.f32 N/A

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

    7. associate-*l/ N/A

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

    8. *-commutative N/A

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

    9. associate-/l/ N/A

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

    10. /-lowering-/.f32 N/A

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

    11. *-lowering-*.f32 N/A

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

    12. *-commutative N/A

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

    13. *-lowering-*.f32 N/A

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

    14. sinh-lowering-sinh.f32 N/A

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

    15. /-lowering-/.f32 N/A

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

    16. /-lowering-/.f32 98.2

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

11. Applied egg-rr 98.2%

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

Alternative 14: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. associate-*l* N/A

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

   7. /-rgt-identity N/A

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

   8. 1-exp N/A

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \frac{v \cdot v}{\color{blue}{e^{0}}}\right)} \]

   9. clear-num N/A

      \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \color{blue}{\frac{1}{\frac{e^{0}}{v \cdot v}}}\right)} \]

   10. 1-exp N/A

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

   11. div-inv N/A

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

   12. *-lowering-*.f32 N/A

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

   13. sinh-lowering-sinh.f32 N/A

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

   14. 1-exp N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\sinh \left(\frac{\color{blue}{e^{0}}}{v}\right) \cdot \frac{2}{\frac{1}{v \cdot v}}} \]

   15. /-lowering-/.f32 N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\sinh \color{blue}{\left(\frac{e^{0}}{v}\right)} \cdot \frac{2}{\frac{1}{v \cdot v}}} \]

   16. 1-exp N/A

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

   17. div-inv N/A

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

   18. 1-exp N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \frac{1}{\frac{\color{blue}{e^{0}}}{v \cdot v}}\right)} \]

   19. clear-num N/A

       \[\leadsto cosTheta\_O \cdot \frac{cosTheta\_i}{\sinh \left(\frac{1}{v}\right) \cdot \left(2 \cdot \color{blue}{\frac{v \cdot v}{e^{0}}}\right)} \]

   20. 1-exp N/A

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

   21. /-rgt-identity N/A

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

9. Applied egg-rr 98.2%

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

Alternative 15: 76.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\frac{1}{v} \cdot \left(cosTheta\_O\_m \cdot \frac{cosTheta\_i\_m}{v}\right)}{2 \cdot \frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* (/ 1.0 v) (* cosTheta_O_m (/ cosTheta_i_m v)))
    (*
     2.0
     (/
      (+
       (/
        (+ 0.008333333333333333 (/ 0.0001984126984126984 (* v v)))
        (* v (* v (* v v))))
       (+ 1.0 (/ 0.16666666666666666 (* v v))))
      v))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((1.0f / v) * (cosTheta_O_m * (cosTheta_i_m / v))) / (2.0f * ((((0.008333333333333333f + (0.0001984126984126984f / (v * v))) / (v * (v * (v * v)))) + (1.0f + (0.16666666666666666f / (v * v)))) / v))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((1.0e0 / v) * (costheta_o_m * (costheta_i_m / v))) / (2.0e0 * ((((0.008333333333333333e0 + (0.0001984126984126984e0 / (v * v))) / (v * (v * (v * v)))) + (1.0e0 + (0.16666666666666666e0 / (v * v)))) / v))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(Float32(1.0) / v) * Float32(cosTheta_O_m * Float32(cosTheta_i_m / v))) / Float32(Float32(2.0) * Float32(Float32(Float32(Float32(Float32(0.008333333333333333) + Float32(Float32(0.0001984126984126984) / Float32(v * v))) / Float32(v * Float32(v * Float32(v * v)))) + Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v)))) / v)))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((single(1.0) / v) * (cosTheta_O_m * (cosTheta_i_m / v))) / (single(2.0) * ((((single(0.008333333333333333) + (single(0.0001984126984126984) / (v * v))) / (v * (v * (v * v)))) + (single(1.0) + (single(0.16666666666666666) / (v * v)))) / v))));
end

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

   1. times-frac N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. 1-exp N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{\color{blue}{e^{0}}}{v}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \color{blue}{\frac{e^{0}}{v}}}{\sinh \left(\frac{1}{v}\right) \cdot 2} \]

   9. 1-exp 98.4

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

9. Applied egg-rr 98.4%

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

10. Taylor expanded in v around -inf

    \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)} \cdot 2} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}\right)\right)} \cdot 2} \]

    2. distribute-frac-neg N/A

       \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\frac{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} - \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)}{v}} \cdot 2} \]

    3. sub-neg N/A

       \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v} \cdot 2} \]

    4. mul-1-neg N/A

       \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}}\right)\right)} + \left(\mathsf{neg}\left(\left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)\right)}{v} \cdot 2} \]

    5. distribute-neg-out N/A

       \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right)}\right)}{v} \cdot 2} \]

    6. remove-double-neg N/A

       \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\frac{\color{blue}{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}}{v} \cdot 2} \]

    7. /-lowering-/.f32 N/A

       \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\frac{\frac{\frac{1}{120} + \frac{1}{5040} \cdot \frac{1}{{v}^{2}}}{{v}^{4}} + \left(1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}\right)}{v}} \cdot 2} \]

12. Simplified 76.2%

    \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot \frac{1}{v}}{\color{blue}{\frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}} \cdot 2} \]

13. Final simplification 76.2%

    \[\leadsto \frac{\frac{1}{v} \cdot \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}{2 \cdot \frac{\frac{0.008333333333333333 + \frac{0.0001984126984126984}{v \cdot v}}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \left(1 + \frac{0.16666666666666666}{v \cdot v}\right)}{v}} \]
14. Add Preprocessing

Alternative 16: 70.3% accurate, 3.0× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (* cosTheta_O_m cosTheta_i_m)
    (/
     (/ 1.0 (* v v))
     (*
      (-
       -1.0
       (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v)))
      (/ -2.0 v)))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m * costheta_i_m) * ((1.0e0 / (v * v)) / (((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))) * ((-2.0e0) / v)))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * cosTheta_i_m) * ((single(1.0) / (v * v)) / ((single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))) * (single(-2.0) / v)))));
end

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

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

10. Simplified 68.8%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2} \]
11. Step-by-step derivation

    1. div-inv N/A

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

    2. *-commutative N/A

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

    3. associate-/l* N/A

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

    4. *-lowering-*.f32 N/A

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

    5. *-lowering-*.f32 N/A

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

    6. /-lowering-/.f32 N/A

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

    7. /-lowering-/.f32 N/A

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

    8. *-lowering-*.f32 N/A

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

    9. associate-*l/ N/A

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

    10. associate-/l* N/A

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

    11. *-lowering-*.f32 N/A

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

12. Applied egg-rr 68.8%

    \[\leadsto \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{\frac{1}{v \cdot v}}{\left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right) \cdot \frac{-2}{v}}} \]
13. Add Preprocessing

Alternative 17: 70.3% accurate, 3.1× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (*
    (/
     cosTheta_i_m
     (*
      (-
       -1.0
       (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v)))
      (/ -2.0 v)))
    (/ cosTheta_O_m (* v v))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_i_m / (((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))) * ((-2.0e0) / v))) * (costheta_o_m / (v * v))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_i_m / ((single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))) * (single(-2.0) / v))) * (cosTheta_O_m / (v * v))));
end

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

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

10. Simplified 68.8%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2} \]
11. Step-by-step derivation

    1. associate-/l/ N/A

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

    2. times-frac N/A

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

    3. *-lowering-*.f32 N/A

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

12. Applied egg-rr 68.8%

    \[\leadsto \color{blue}{\frac{cosTheta\_i}{\left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right) \cdot \frac{-2}{v}} \cdot \frac{cosTheta\_O}{v \cdot v}} \]
13. Add Preprocessing

Alternative 18: 70.3% accurate, 3.4× speedup

Math

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

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* cosTheta_O_m cosTheta_i_m)
    (*
     (* v v)
     (*
      (-
       -1.0
       (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v)))
      (/ -2.0 v)))))))

C

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

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * ((costheta_o_m * costheta_i_m) / ((v * v) * (((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))) * ((-2.0e0) / v)))))
end function

Julia

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

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * ((cosTheta_O_m * cosTheta_i_m) / ((v * v) * ((single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))) * (single(-2.0) / v)))));
end

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

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

10. Simplified 68.8%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2} \]
11. Step-by-step derivation

    1. associate-/l/ N/A

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

    2. *-commutative N/A

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

    3. /-lowering-/.f32 N/A

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

    4. *-lowering-*.f32 N/A

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

    5. *-commutative N/A

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

    6. *-lowering-*.f32 N/A

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

    7. *-lowering-*.f32 N/A

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

    8. associate-*l/ N/A

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

    9. associate-/l* N/A

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

    10. *-lowering-*.f32 N/A

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

12. Applied egg-rr 68.8%

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{\left(v \cdot v\right) \cdot \left(\left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right) \cdot \frac{-2}{v}\right)}} \]
13. Add Preprocessing

Alternative 19: 70.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} cosTheta_i\_m = \left|cosTheta\_i\right| \\ cosTheta_i\_s = \mathsf{copysign}\left(1, cosTheta\_i\right) \\ cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(cosTheta\_i\_s \cdot \frac{\left(cosTheta\_O\_m \cdot cosTheta\_i\_m\right) \cdot -0.5}{v \cdot \left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right)}\right) \end{array} \]

FPCore

cosTheta_i\_m = (fabs.f32 cosTheta_i)
cosTheta_i\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_i)
cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i_s cosTheta_i_m cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (*
   cosTheta_i_s
   (/
    (* (* cosTheta_O_m cosTheta_i_m) -0.5)
    (*
     v
     (-
      -1.0
      (/
       (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v)))
       (* v v))))))))

C

cosTheta_i\_m = fabs(cosTheta_i);
cosTheta_i\_s = copysign(1.0, cosTheta_i);
cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i_m < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i_s, float cosTheta_i_m, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (cosTheta_i_s * (((cosTheta_O_m * cosTheta_i_m) * -0.5f) / (v * (-1.0f - ((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v))))));
}

Fortran

cosTheta_i\_m = abs(costheta_i)
cosTheta_i\_s = copysign(1.0d0, costheta_i)
cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i_s, costheta_i_m, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (costheta_i_s * (((costheta_o_m * costheta_i_m) * (-0.5e0)) / (v * ((-1.0e0) - ((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / (v * v))))))
end function

Julia

cosTheta_i\_m = abs(cosTheta_i)
cosTheta_i\_s = copysign(1.0, cosTheta_i)
cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(cosTheta_i_s * Float32(Float32(Float32(cosTheta_O_m * cosTheta_i_m) * Float32(-0.5)) / Float32(v * Float32(Float32(-1.0) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)))))))
end

MATLAB

cosTheta_i\_m = abs(cosTheta_i);
cosTheta_i\_s = sign(cosTheta_i) * abs(1.0);
cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i_s, cosTheta_i_m, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (cosTheta_i_s * (((cosTheta_O_m * cosTheta_i_m) * single(-0.5)) / (v * (single(-1.0) - ((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / (v * v))))));
end

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

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

10. Simplified 68.8%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2} \]

11. Taylor expanded in cosTheta_i around 0

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

    1. associate-*r/ N/A

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

    2. /-lowering-/.f32 N/A

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

    3. *-lowering-*.f32 N/A

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

    4. *-lowering-*.f32 N/A

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

    5. *-lowering-*.f32 N/A

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

    8. +-commutative N/A

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

    9. mul-1-neg N/A

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

    10. unsub-neg N/A

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

    11. --lowering--.f32 N/A

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

    12. /-lowering-/.f32 N/A

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

13. Simplified 68.8%

    \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)}{v \cdot \left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right)}} \]

14. Final simplification 68.8%

    \[\leadsto \frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot -0.5}{v \cdot \left(-1 - \frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v}\right)} \]
15. Add Preprocessing

Alternative 20: 64.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i \cdot cosTheta\_O}{e^{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, 0\right)} \cdot \left(v \cdot v\right)}}{\sinh \left(\frac{1}{v}\right) \cdot 2}} \]

5. Taylor expanded in sinTheta_i around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f32 98.1

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

7. Simplified 98.1%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f32 N/A

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

10. Simplified 68.8%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot v}}{\color{blue}{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{-v \cdot v} + -1}{-v}} \cdot 2} \]

11. Taylor expanded in v around inf

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

    1. /-lowering-/.f32 N/A

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

    2. +-lowering-+.f32 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. /-lowering-/.f32 N/A

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

    6. unpow2 N/A

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

    7. *-lowering-*.f32 62.0

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

13. Simplified 62.0%

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

14. Final simplification 62.0%

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

Alternative 21: 59.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. /-lowering-/.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. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

5. Simplified 55.7%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

   5. 1-exp N/A

      \[\leadsto \frac{\frac{\color{blue}{e^{0}}}{\frac{2}{cosTheta\_i \cdot cosTheta\_O}}}{v} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{0}}{\frac{2}{cosTheta\_i \cdot cosTheta\_O}}}}{v} \]

   7. 1-exp N/A

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

   8. /-lowering-/.f32 N/A

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

   9. *-lowering-*.f32 56.2

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

7. Applied egg-rr 56.2%

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

8. Final simplification 56.2%

   \[\leadsto \frac{\frac{1}{\frac{2}{cosTheta\_O \cdot cosTheta\_i}}}{v} \]
9. Add Preprocessing

Alternative 22: 58.9% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. /-lowering-/.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. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

5. Simplified 55.7%

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

   1. clear-num N/A

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

   2. 1-exp N/A

      \[\leadsto \frac{\color{blue}{e^{0}}}{\frac{v}{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{2}}} \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{0}}{\frac{v}{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot \frac{1}{2}}}} \]

   4. 1-exp N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 56.2

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

7. Applied egg-rr 56.2%

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

8. Final simplification 56.2%

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

Alternative 23: 58.9% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. /-lowering-/.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. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

5. Simplified 55.7%

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

   1. clear-num N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. /-lowering-/.f32 N/A

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

   6. /-lowering-/.f32 N/A

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

   7. *-lowering-*.f32 56.1

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

7. Applied egg-rr 56.1%

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

8. Final simplification 56.1%

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

Alternative 24: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. /-lowering-/.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. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

5. Simplified 55.7%

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

6. Final simplification 55.7%

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

Alternative 25: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. /-lowering-/.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. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

5. Simplified 55.7%

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

7. Applied egg-rr 55.7%

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

8. Final simplification 55.7%

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

Alternative 26: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. /-lowering-/.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. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

5. Simplified 55.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. *-lowering-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. clear-num N/A

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

   11. /-lowering-/.f32 N/A

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

   12. *-lowering-*.f32 55.6

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

7. Applied egg-rr 55.6%

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

8. Final simplification 55.6%

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

Alternative 27: 58.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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. /-lowering-/.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. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 55.7

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

5. Simplified 55.7%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. *-lowering-*.f32 55.6

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

7. Applied egg-rr 55.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. *-lowering-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. clear-num N/A

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

   10. /-lowering-/.f32 55.6

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

9. Applied egg-rr 55.6%

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

10. Final simplification 55.6%

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

Reproduce

herbie shell --seed 2024197 
(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)))