HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 22.2s

Alternatives: 18

Speedup: 1.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i (* cosTheta_O (/ 0.5 v)))
  (*
   (/ 1.0 v)
   (/ (exp (/ (* sinTheta_i sinTheta_O) (- v))) (sinh (/ 1.0 v))))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * (cosTheta_O * (0.5f / v))) * ((1.0f / v) * (expf(((sinTheta_i * sinTheta_O) / -v)) / sinhf((1.0f / v))));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (costheta_i * (costheta_o * (0.5e0 / v))) * ((1.0e0 / v) * (exp(((sintheta_i * sintheta_o) / -v)) / sinh((1.0e0 / v))))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * Float32(cosTheta_O * Float32(Float32(0.5) / v))) * Float32(Float32(Float32(1.0) / v) * Float32(exp(Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v))) / sinh(Float32(Float32(1.0) / v)))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * (cosTheta_O * (single(0.5) / v))) * ((single(1.0) / v) * (exp(((sinTheta_i * sinTheta_O) / -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

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-rgt-identity N/A

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

   9. lift-/.f32 N/A

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

   10. lift-sinh.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. remove-double-neg N/A

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

   13. frac-times N/A

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

   14. un-div-inv N/A

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

4. Applied egg-rr 98.8%

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

   1. associate-*r/ N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 98.9

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

6. Applied egg-rr 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (* cosTheta_i 0.5) (* v (sinh (/ 1.0 v))))
  (* cosTheta_O (/ (exp (* sinTheta_i (/ sinTheta_O (- v)))) v))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * 0.5f) / (v * sinhf((1.0f / v)))) * (cosTheta_O * (expf((sinTheta_i * (sinTheta_O / -v))) / v));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = ((costheta_i * 0.5e0) / (v * sinh((1.0e0 / v)))) * (costheta_o * (exp((sintheta_i * (sintheta_o / -v))) / v))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * Float32(0.5)) / Float32(v * sinh(Float32(Float32(1.0) / v)))) * Float32(cosTheta_O * Float32(exp(Float32(sinTheta_i * Float32(sinTheta_O / Float32(-v)))) / v)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * single(0.5)) / (v * sinh((single(1.0) / v)))) * (cosTheta_O * (exp((sinTheta_i * (sinTheta_O / -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

4. Applied egg-rr 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f32 N/A

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

6. Applied egg-rr 98.9%

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

Alternative 3: 98.7% accurate, 1.2× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (fma
   sinTheta_i
   (/
    (*
     (* cosTheta_i (* cosTheta_O 0.5))
     (* sinTheta_i (* sinTheta_O sinTheta_O)))
    (* v (* v v)))
   (*
    (fma (- cosTheta_i) (/ (* sinTheta_i sinTheta_O) v) cosTheta_i)
    (/ cosTheta_O v)))
  (* (/ 0.5 v) (/ 1.0 (sinh (/ 1.0 v))))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return fmaf(sinTheta_i, (((cosTheta_i * (cosTheta_O * 0.5f)) * (sinTheta_i * (sinTheta_O * sinTheta_O))) / (v * (v * v))), (fmaf(-cosTheta_i, ((sinTheta_i * sinTheta_O) / v), cosTheta_i) * (cosTheta_O / v))) * ((0.5f / v) * (1.0f / sinhf((1.0f / v))));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(fma(sinTheta_i, Float32(Float32(Float32(cosTheta_i * Float32(cosTheta_O * Float32(0.5))) * Float32(sinTheta_i * Float32(sinTheta_O * sinTheta_O))) / Float32(v * Float32(v * v))), Float32(fma(Float32(-cosTheta_i), Float32(Float32(sinTheta_i * sinTheta_O) / v), cosTheta_i) * Float32(cosTheta_O / v))) * Float32(Float32(Float32(0.5) / v) * Float32(Float32(1.0) / sinh(Float32(Float32(1.0) / v)))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

5. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{0.5 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)\right)}{v \cdot \left(v \cdot v\right)}, \frac{cosTheta\_O \cdot \mathsf{fma}\left(-cosTheta\_i, \frac{sinTheta\_O \cdot sinTheta\_i}{v}, cosTheta\_i\right)}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{\left(cosTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)\right) \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)}{v \cdot \left(v \cdot v\right)}, \mathsf{fma}\left(-cosTheta\_i, \frac{sinTheta\_i \cdot sinTheta\_O}{v}, cosTheta\_i\right) \cdot \frac{cosTheta\_O}{v}\right) \cdot \frac{1}{\sinh \left(\frac{1}{v}\right) \cdot \left(v \cdot 2\right)}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. div-inv N/A

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

   6. lift-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. clear-num N/A

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

   10. lift-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 98.8

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{\mathsf{fma}\left(\frac{1}{v}, cosTheta\_O \cdot \mathsf{fma}\left(-cosTheta\_i, \frac{sinTheta\_i \cdot sinTheta\_O}{v}, cosTheta\_i\right), \frac{\left(sinTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)\right) \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (fma
   (/ 1.0 v)
   (*
    cosTheta_O
    (fma (- cosTheta_i) (/ (* sinTheta_i sinTheta_O) v) cosTheta_i))
   (/
    (*
     (* sinTheta_i (* cosTheta_O 0.5))
     (* cosTheta_i (* sinTheta_i (* sinTheta_O sinTheta_O))))
    (* v (* v v))))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return fmaf((1.0f / v), (cosTheta_O * fmaf(-cosTheta_i, ((sinTheta_i * sinTheta_O) / v), cosTheta_i)), (((sinTheta_i * (cosTheta_O * 0.5f)) * (cosTheta_i * (sinTheta_i * (sinTheta_O * sinTheta_O)))) / (v * (v * v)))) / (v * (sinhf((1.0f / v)) * 2.0f));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(fma(Float32(Float32(1.0) / v), Float32(cosTheta_O * fma(Float32(-cosTheta_i), Float32(Float32(sinTheta_i * sinTheta_O) / v), cosTheta_i)), Float32(Float32(Float32(sinTheta_i * Float32(cosTheta_O * Float32(0.5))) * Float32(cosTheta_i * Float32(sinTheta_i * Float32(sinTheta_O * sinTheta_O)))) / Float32(v * Float32(v * v)))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

5. Simplified 98.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{0.5 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)\right)}{v \cdot \left(v \cdot v\right)}, \frac{cosTheta\_O \cdot \mathsf{fma}\left(-cosTheta\_i, \frac{sinTheta\_O \cdot sinTheta\_i}{v}, cosTheta\_i\right)}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Applied egg-rr 98.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{v}, cosTheta\_O \cdot \mathsf{fma}\left(-cosTheta\_i, \frac{sinTheta\_i \cdot sinTheta\_O}{v}, cosTheta\_i\right), \frac{\left(sinTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)\right) \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

8. Final simplification 98.9%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{v}, cosTheta\_O \cdot \mathsf{fma}\left(-cosTheta\_i, \frac{sinTheta\_i \cdot sinTheta\_O}{v}, cosTheta\_i\right), \frac{\left(sinTheta\_i \cdot \left(cosTheta\_O \cdot 0.5\right)\right) \cdot \left(cosTheta\_i \cdot \left(sinTheta\_i \cdot \left(sinTheta\_O \cdot sinTheta\_O\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\right)}{v \cdot \left(\sinh \left(\frac{1}{v}\right) \cdot 2\right)} \]
9. Add Preprocessing

Alternative 5: 98.7% accurate, 1.5× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   cosTheta_O
   (* cosTheta_i (fma sinTheta_O (/ (- sinTheta_i) (* v v)) (/ 1.0 v))))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(cosTheta_i * fma(sinTheta_O, Float32(Float32(-sinTheta_i) / Float32(v * v)), Float32(Float32(1.0) / v)))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

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

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

   4. mul-1-neg N/A

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

   5. lower-fma.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 98.8

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 6: 98.5% accurate, 1.6× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (fma sinTheta_i (/ sinTheta_O (- v)) 1.0) (/ (* cosTheta_i cosTheta_O) v))
  (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(fma(sinTheta_i, Float32(sinTheta_O / Float32(-v)), Float32(1.0)) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.7

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 7: 98.5% accurate, 1.6× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ (* cosTheta_i 0.5) (* v (sinh (/ 1.0 v))))
  (/ (fma cosTheta_O (/ (* sinTheta_i sinTheta_O) (- v)) cosTheta_O) v)))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * 0.5f) / (v * sinhf((1.0f / v)))) * (fmaf(cosTheta_O, ((sinTheta_i * sinTheta_O) / -v), cosTheta_O) / v);
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * Float32(0.5)) / Float32(v * sinh(Float32(Float32(1.0) / v)))) * Float32(fma(cosTheta_O, Float32(Float32(sinTheta_i * sinTheta_O) / Float32(-v)), cosTheta_O) / 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

4. Applied egg-rr 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f32 N/A

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f32 N/A

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

   10. mul-1-neg N/A

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

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

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

   12. mul-1-neg N/A

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

   13. lower-*.f32 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f32 98.7

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 8: 98.5% accurate, 1.6× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* (fma sinTheta_i (/ sinTheta_O (- v)) 1.0) (/ (* cosTheta_i cosTheta_O) v))
  (/ 0.5 (* v (sinh (/ 1.0 v))))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (fmaf(sinTheta_i, (sinTheta_O / -v), 1.0f) * ((cosTheta_i * cosTheta_O) / v)) * (0.5f / (v * sinhf((1.0f / v))));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(fma(sinTheta_i, Float32(sinTheta_O / Float32(-v)), Float32(1.0)) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) * Float32(Float32(0.5) / Float32(v * sinh(Float32(Float32(1.0) / v)))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.7

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

5. Simplified 98.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f32 N/A

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 9: 98.5% accurate, 1.6× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (fma sinTheta_i (/ sinTheta_O (- v)) 1.0) (* cosTheta_i cosTheta_O))
  (* v (* v (* (sinh (/ 1.0 v)) 2.0)))))

C

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

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(fma(sinTheta_i, Float32(sinTheta_O / Float32(-v)), Float32(1.0)) * Float32(cosTheta_i * cosTheta_O)) / Float32(v * Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.7

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

5. Simplified 98.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-sinh.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*r/ N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f32 N/A

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 10: 98.5% accurate, 1.8× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_O (* cosTheta_i (/ 1.0 v))) (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (costheta_o * (costheta_i * (1.0e0 / v))) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(cosTheta_i * Float32(Float32(1.0) / v))) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * (cosTheta_i * (single(1.0) / v))) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.6

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 11: 98.3% accurate, 1.8× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* cosTheta_O (/ cosTheta_i v)) (* v (* (sinh (/ 1.0 v)) 2.0))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = (costheta_o * (costheta_i / v)) / (v * (sinh((1.0e0 / v)) * 2.0e0))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) / Float32(v * Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_O * (cosTheta_i / v)) / (v * (sinh((single(1.0) / v)) * single(2.0)));
end

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.5

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 12: 98.4% accurate, 1.8× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ (* cosTheta_i 0.5) (* v (sinh (/ 1.0 v)))) (/ cosTheta_O v)))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_i * 0.5f) / (v * sinhf((1.0f / v)))) * (cosTheta_O / v);
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = ((costheta_i * 0.5e0) / (v * sinh((1.0e0 / v)))) * (costheta_o / v)
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_i * Float32(0.5)) / Float32(v * sinh(Float32(Float32(1.0) / v)))) * Float32(cosTheta_O / v))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_i * single(0.5)) / (v * sinh((single(1.0) / v)))) * (cosTheta_O / 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

4. Applied egg-rr 98.6%

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

   1. lift-/.f32 N/A

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

   2. lift-sinh.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f32 N/A

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.4

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

9. Simplified 98.4%

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

Alternative 13: 69.4% accurate, 2.5× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (fma sinTheta_i (/ sinTheta_O (- v)) 1.0) (/ (* cosTheta_i cosTheta_O) v))
  (*
   v
   (*
    2.0
    (/
     (+
      (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) (* v v))
      1.0)
     v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (fmaf(sinTheta_i, (sinTheta_O / -v), 1.0f) * ((cosTheta_i * cosTheta_O) / v)) / (v * (2.0f * ((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / (v * v)) + 1.0f) / v)));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(fma(sinTheta_i, Float32(sinTheta_O / Float32(-v)), Float32(1.0)) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(v * Float32(Float32(2.0) * Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / Float32(v * v)) + Float32(1.0)) / v))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f32 N/A

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

8. Simplified 72.7%

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

9. Final simplification 72.7%

   \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{-v}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{v \cdot \left(2 \cdot \frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v \cdot v} + 1}{v}\right)} \]
10. Add Preprocessing

Alternative 14: 63.3% accurate, 3.8× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (fma sinTheta_i (/ sinTheta_O (- v)) 1.0) (/ (* cosTheta_i cosTheta_O) v))
  (+ 2.0 (/ 0.3333333333333333 (* v v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (fmaf(sinTheta_i, (sinTheta_O / -v), 1.0f) * ((cosTheta_i * cosTheta_O) / v)) / (2.0f + (0.3333333333333333f / (v * v)));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(fma(sinTheta_i, Float32(sinTheta_O / Float32(-v)), Float32(1.0)) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(2.0) + Float32(Float32(0.3333333333333333) / Float32(v * v))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-neg.f32 98.7

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

5. Simplified 98.7%

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

6. Taylor expanded in v around inf

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

   1. lower-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f32 N/A

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

   5. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{\mathsf{neg}\left(v\right)}, 1\right) \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{2 + \frac{\frac{1}{3}}{\color{blue}{v \cdot v}}} \]

   6. lower-*.f32 66.4

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

8. Simplified 66.4%

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

Alternative 15: 58.0% accurate, 8.2× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ v (* cosTheta_i (* cosTheta_O 0.5)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / (cosTheta_i * (cosTheta_O * 0.5f)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = 1.0e0 / (v / (costheta_i * (costheta_o * 0.5e0)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(v / Float32(cosTheta_i * Float32(cosTheta_O * Float32(0.5)))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / (cosTheta_i * (cosTheta_O * single(0.5))));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.6

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

5. Simplified 60.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 61.7

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 61.7

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

7. Applied egg-rr 61.7%

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

Alternative 16: 57.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ 0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (* cosTheta_i cosTheta_O) v)))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((cosTheta_i * cosTheta_O) / v);
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = 0.5e0 * ((costheta_i * costheta_o) / v)
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(Float32(cosTheta_i * cosTheta_O) / v))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_i * cosTheta_O) / v);
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.6

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

5. Simplified 60.6%

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

   1. lift-*.f32 N/A

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

   2. associate-*l/ N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 60.6

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. lift-*.f32 N/A

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

   12. lower-/.f32 60.7

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

7. Applied egg-rr 60.7%

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

8. Final simplification 60.7%

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

Alternative 17: 57.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_i \cdot \frac{cosTheta\_O}{v \cdot 2} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_i (/ cosTheta_O (* v 2.0))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_i * (cosTheta_O / (v * 2.0f));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = costheta_i * (costheta_o / (v * 2.0e0))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_i * Float32(cosTheta_O / Float32(v * Float32(2.0))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_i * (cosTheta_O / (v * single(2.0)));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.6

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

5. Simplified 60.6%

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

   1. lift-*.f32 N/A

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

   2. associate-/l* N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 60.6

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

7. Applied egg-rr 60.6%

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

   1. lift-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. lower-*.f32 60.6

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. clear-num N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 60.6

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

9. Applied egg-rr 60.6%

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

10. Final simplification 60.6%

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

Alternative 18: 57.5% accurate, 12.4× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O (* cosTheta_i (/ 0.5 v))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O * (cosTheta_i * (0.5f / v));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 = costheta_o * (costheta_i * (0.5e0 / v))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O * Float32(cosTheta_i * Float32(Float32(0.5) / v)))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O * (cosTheta_i * (single(0.5) / v));
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in v around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 60.6

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

5. Simplified 60.6%

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

   1. lift-*.f32 N/A

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

   2. associate-/l* N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 60.6

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

7. Applied egg-rr 60.6%

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

Reproduce

herbie shell --seed 2024211 
(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)))