HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.9%

Time: 14.8s

Alternatives: 15

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. remove-double-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

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

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

   10. associate-*l* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f32 99.0

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 99.0

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. remove-double-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

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

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

   10. associate-*l* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 98.9

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 3: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. remove-double-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

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

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

   10. associate-*l* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. lower-*.f32 98.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

7. Applied rewrites 98.9%

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

Alternative 5: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. lift-/.f32 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.8%

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

Alternative 6: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. lift-/.f32 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

9. Applied rewrites 98.7%

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

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. lift-/.f32 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.7

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

9. Applied rewrites 98.7%

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

Alternative 8: 70.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{cosTheta\_O\_m \cdot cosTheta\_i}{\frac{-2 - \frac{\frac{\frac{0.016666666666666666}{v \cdot v} + 0.3333333333333333}{v}}{v}}{-v} \cdot \left(v \cdot v\right)} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (* cosTheta_O_m cosTheta_i)
   (*
    (/
     (-
      -2.0
      (/ (/ (+ (/ 0.016666666666666666 (* v v)) 0.3333333333333333) v) v))
     (- v))
    (* v v)))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (((-2.0f - ((((0.016666666666666666f / (v * v)) + 0.3333333333333333f) / v) / v)) / -v) * (v * v)));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * costheta_i) / ((((-2.0e0) - ((((0.016666666666666666e0 / (v * v)) + 0.3333333333333333e0) / v) / v)) / -v) * (v * v)))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(cosTheta_O_m * cosTheta_i) / Float32(Float32(Float32(Float32(-2.0) - Float32(Float32(Float32(Float32(Float32(0.016666666666666666) / Float32(v * v)) + Float32(0.3333333333333333)) / v) / v)) / Float32(-v)) * Float32(v * v))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (((single(-2.0) - ((((single(0.016666666666666666) / (v * v)) + single(0.3333333333333333)) / v) / v)) / -v) * (v * v)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. lift-/.f32 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in v around -inf

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{v}\right) \cdot \left(\color{blue}{v} \cdot v\right)} \]
9. Step-by-step derivation

10. Applied rewrites 68.6%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\frac{-2 - \frac{\frac{\frac{0.016666666666666666}{v \cdot v} + 0.3333333333333333}{v}}{v}}{-v} \cdot \left(\color{blue}{v} \cdot v\right)} \]
11. Add Preprocessing

Alternative 9: 70.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \frac{cosTheta\_O\_m \cdot cosTheta\_i}{\left(-v\right) \cdot \left(-2 - \frac{\frac{\frac{0.016666666666666666}{v \cdot v} + 0.3333333333333333}{v}}{v}\right)} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (* cosTheta_O_m cosTheta_i)
   (*
    (- v)
    (-
     -2.0
     (/ (/ (+ (/ 0.016666666666666666 (* v v)) 0.3333333333333333) v) v))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (-v * (-2.0f - ((((0.016666666666666666f / (v * v)) + 0.3333333333333333f) / v) / v))));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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_o_m * costheta_i) / (-v * ((-2.0e0) - ((((0.016666666666666666e0 / (v * v)) + 0.3333333333333333e0) / v) / v))))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(Float32(cosTheta_O_m * cosTheta_i) / Float32(Float32(-v) * Float32(Float32(-2.0) - Float32(Float32(Float32(Float32(Float32(0.016666666666666666) / Float32(v * v)) + Float32(0.3333333333333333)) / v) / v)))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * ((cosTheta_O_m * cosTheta_i) / (-v * (single(-2.0) - ((((single(0.016666666666666666) / (v * v)) + single(0.3333333333333333)) / v) / v))));
end

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. lift-/.f32 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in v around -inf

   \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{-1 \cdot \color{blue}{\left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 68.6%

    \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{\left(-v\right) \cdot \color{blue}{\left(-2 - \frac{\frac{\frac{0.016666666666666666}{v \cdot v} + 0.3333333333333333}{v}}{v}\right)}} \]
11. Add Preprocessing

Alternative 10: 64.4% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. exp-neg N/A

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

   7. lift-/.f32 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. rec-exp N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 98.7

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

7. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

9. Applied rewrites 98.7%

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

10. Taylor expanded in v around inf

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

12. Applied rewrites 61.6%

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

13. Final simplification 61.6%

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

Alternative 11: 59.1% accurate, 8.2× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, 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 cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (/ 1.0 (/ (* 2.0 v) (* cosTheta_i cosTheta_O_m)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, 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, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    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 * (1.0e0 / ((2.0e0 * v) / (costheta_i * costheta_o_m)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 54.9%

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

9. Applied rewrites 55.8%

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

Alternative 12: 59.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 55.8%

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

Alternative 13: 59.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 55.7%

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

Alternative 14: 58.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 54.9%

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

Alternative 15: 58.7% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 54.9%

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

9. Applied rewrites 54.9%

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

Reproduce

herbie shell --seed 2024324 
(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)))