HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.7%

Time: 21.0s

Alternatives: 22

Speedup: 1.7×

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: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. times-frac N/A

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

4. Applied egg-rr 99.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. lift-/.f32 N/A

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

   4. lower-*.f32 99.1

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 99.1

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

9. Simplified 99.1%

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

Alternative 2: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. times-frac N/A

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

4. Applied egg-rr 99.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. lift-/.f32 N/A

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

   4. lower-*.f32 99.1

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in v around inf

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 99.1

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

9. Simplified 99.1%

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

Alternative 3: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. times-frac N/A

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

4. Applied egg-rr 99.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. lift-/.f32 N/A

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

   4. lower-*.f32 99.1

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 98.9

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

9. Simplified 98.9%

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

Alternative 4: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. neg-mul-1 N/A

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

   5. lift-exp.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. lift-/.f32 N/A

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

   8. div-inv N/A

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

   9. lift-/.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. times-frac N/A

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

   14. associate-*l/ N/A

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

   15. lower-/.f32 N/A

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

9. Applied egg-rr 98.4%

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

    1. lift-*.f32 N/A

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

    2. associate-*r/ N/A

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

    3. lift-*.f32 N/A

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

    4. times-frac N/A

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

    5. lower-*.f32 N/A

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

    6. lower-/.f32 N/A

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

    7. lower-/.f32 98.7

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

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

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

Alternative 5: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

   1. *-commutative N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. neg-mul-1 N/A

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

   6. lift-exp.f32 N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. lift-exp.f32 N/A

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

   12. lift--.f32 N/A

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

   13. associate-*r* N/A

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

   14. times-frac N/A

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

   15. un-div-inv N/A

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

   16. lift-/.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 N/A

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 6: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \begin{array}{l} \mathbf{if}\;v \leq 0.33500000834465027:\\ \;\;\;\;\frac{cosTheta\_O\_m \cdot cosTheta\_i}{\left(v \cdot v\right) \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O\_m \cdot cosTheta\_i}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)}\\ \end{array} \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (if (<= v 0.33500000834465027)
    (/ (* cosTheta_O_m cosTheta_i) (* (* v v) (+ (exp (/ 1.0 v)) -1.0)))
    (/
     (* cosTheta_O_m cosTheta_i)
     (*
      v
      (-
       (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
       -2.0))))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (v <= 0.33500000834465027f) {
		tmp = (cosTheta_O_m * cosTheta_i) / ((v * v) * (expf((1.0f / v)) + -1.0f));
	} else {
		tmp = (cosTheta_O_m * cosTheta_i) / (v * (((0.3333333333333333f + (0.016666666666666666f / (v * v))) / (v * v)) - -2.0f));
	}
	return cosTheta_O_s * tmp;
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (v <= 0.33500000834465027e0) then
        tmp = (costheta_o_m * costheta_i) / ((v * v) * (exp((1.0e0 / v)) + (-1.0e0)))
    else
        tmp = (costheta_o_m * costheta_i) / (v * (((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0)))
    end if
    code = costheta_o_s * tmp
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (v <= Float32(0.33500000834465027))
		tmp = Float32(Float32(cosTheta_O_m * cosTheta_i) / Float32(Float32(v * v) * Float32(exp(Float32(Float32(1.0) / v)) + Float32(-1.0))));
	else
		tmp = Float32(Float32(cosTheta_O_m * cosTheta_i) / Float32(v * Float32(Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(0.016666666666666666) / Float32(v * v))) / Float32(v * v)) - Float32(-2.0))));
	end
	return Float32(cosTheta_O_s * tmp)
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp_2 = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (v <= single(0.33500000834465027))
		tmp = (cosTheta_O_m * cosTheta_i) / ((v * v) * (exp((single(1.0) / v)) + single(-1.0)));
	else
		tmp = (cosTheta_O_m * cosTheta_i) / (v * (((single(0.3333333333333333) + (single(0.016666666666666666) / (v * v))) / (v * v)) - single(-2.0)));
	end
	tmp_2 = cosTheta_O_s * tmp;
end

Derivation

1. Split input into 2 regimes

2. if v < 0.335000008

   1. Initial program 98.1%

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

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

      8. lift-sinh.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. times-frac N/A

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in sinTheta_i around 0

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

      1. lower--.f32 N/A

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

      2. lower-exp.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. rec-exp N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 97.9

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

   7. Simplified 97.9%

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

   8. Taylor expanded in v around inf

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

   10. Simplified 75.1%

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

   if 0.335000008 < v

   1. Initial program 99.0%

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

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

      8. lift-sinh.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. times-frac N/A

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in sinTheta_i around 0

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

      1. lower--.f32 N/A

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

      2. lower-exp.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. rec-exp N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 98.9

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

   7. Simplified 98.9%

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

   8. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\mathsf{neg}\left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}} \]

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

         \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-*.f32 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)}} \]

      6. lower-neg.f32 N/A

         \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)}} \]

      7. sub-neg N/A

         \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right)}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) + -1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(2\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)}\right)\right)\right)} \]

      10. unsub-neg N/A

          \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) - \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)} \]

      11. lower--.f32 N/A

          \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) - \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\left(\color{blue}{-2} - \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)\right)} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\left(-2 - \color{blue}{\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}}\right)\right)\right)} \]

   10. Simplified 79.3%

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(-\left(-2 - \frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.33500000834465027:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{\left(v \cdot v\right) \cdot \left(e^{\frac{1}{v}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v \cdot \left(\frac{0.3333333333333333 + \frac{0.016666666666666666}{v \cdot v}}{v \cdot v} - -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. neg-mul-1 N/A

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

   5. lift-exp.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. lift-/.f32 N/A

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

   8. div-inv N/A

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

   9. lift-/.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-/l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 8: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

   1. *-commutative N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. neg-mul-1 N/A

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

   6. lift-exp.f32 N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift-/.f32 N/A

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

   11. lift-exp.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-/l* N/A

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

   16. lower-*.f32 N/A

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 9: 70.0% accurate, 4.5× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (/
   (* cosTheta_O_m cosTheta_i)
   (*
    v
    (-
     (/ (+ 0.3333333333333333 (/ 0.016666666666666666 (* v v))) (* v v))
     -2.0)))))

C

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

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * ((costheta_o_m * costheta_i) / (v * (((0.3333333333333333e0 + (0.016666666666666666e0 / (v * v))) / (v * v)) - (-2.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in v around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{\mathsf{neg}\left(v \cdot \left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)}} \]

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

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{\color{blue}{v \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)}} \]

   3. mul-1-neg N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)}} \]

   6. lower-neg.f32 N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2\right)\right)\right)}} \]

   7. sub-neg N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right)}\right)\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) + -1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(2\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)}\right)\right)\right)} \]

   10. unsub-neg N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) - \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)} \]

   11. lower--.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) - \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\left(\color{blue}{-2} - \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}\right)\right)\right)} \]

   13. lower-/.f32 N/A

       \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v \cdot \left(\mathsf{neg}\left(\left(-2 - \color{blue}{\frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}}}\right)\right)\right)} \]

10. Simplified 70.8%

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

11. Final simplification 70.8%

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

Alternative 10: 63.8% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.0

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

10. Simplified 65.0%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-+.f32 N/A

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

    4. times-frac N/A

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

    5. associate-*l/ N/A

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

    6. lower-/.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-/.f32 65.0

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

12. Applied egg-rr 65.0%

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

Alternative 11: 63.8% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.0

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

10. Simplified 65.0%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-+.f32 N/A

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

    4. times-frac N/A

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

    5. lower-*.f32 N/A

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

    6. lower-/.f32 N/A

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

    7. lower-/.f32 65.0

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

12. Applied egg-rr 65.0%

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

Alternative 12: 63.8% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.0

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

10. Simplified 65.0%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-+.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. associate-/l* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

12. Applied egg-rr 65.0%

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

13. Final simplification 65.0%

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

Alternative 13: 63.8% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.0

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

10. Simplified 65.0%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-+.f32 N/A

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

    4. *-commutative N/A

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

    5. lift-*.f32 N/A

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

    6. associate-/l* N/A

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

    7. lower-*.f32 N/A

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

    8. lower-/.f32 65.0

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

    9. lift-*.f32 N/A

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

    10. lift-+.f32 N/A

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

    11. distribute-rgt-in N/A

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

    12. *-commutative N/A

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

    13. lift-/.f32 N/A

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

    14. div-inv N/A

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

    15. associate-*l* N/A

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

    16. lift-*.f32 N/A

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

    17. pow2 N/A

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

    18. pow-flip N/A

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

    19. pow-plus N/A

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

    20. metadata-eval N/A

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

    21. metadata-eval N/A

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

    22. inv-pow N/A

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

12. Applied egg-rr 65.0%

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

Alternative 14: 58.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 59.5

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 59.5

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

7. Applied egg-rr 59.5%

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

   1. metadata-eval N/A

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

   2. associate-/r* N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. un-div-inv N/A

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

   9. clear-num N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 59.8

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 59.8

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

9. Applied egg-rr 59.8%

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

Alternative 15: 58.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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 59.8

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

   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. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 59.8

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 59.8

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

7. Applied egg-rr 59.8%

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

Alternative 16: 58.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 59.8

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 59.8

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

7. Applied egg-rr 59.8%

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

Alternative 17: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

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

6. Final simplification 59.5%

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

Alternative 18: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 59.5

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

   14. lift-*.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. associate-*r/ N/A

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

   17. *-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. lower-/.f32 59.5

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

   20. lift-*.f32 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f32 59.5

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

7. Applied egg-rr 59.5%

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

8. Final simplification 59.5%

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

Alternative 19: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 59.5

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 59.5

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

7. Applied egg-rr 59.5%

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

Alternative 20: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 59.5

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 59.5

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

7. Applied egg-rr 59.5%

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

   1. lift-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lift-/.f32 59.4

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

9. Applied egg-rr 59.4%

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

10. Final simplification 59.4%

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

Alternative 21: 58.2% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
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 59.5

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

5. Simplified 59.5%

   \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 0.5}{v}} \]
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. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 59.5

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 59.5

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

7. Applied egg-rr 59.5%

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

   1. lift-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lift-/.f32 59.4

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

9. Applied egg-rr 59.4%

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

    1. lift-/.f32 N/A

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

    2. associate-*l* N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f32 N/A

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

    6. lower-*.f32 59.4

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

11. Applied egg-rr 59.4%

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

12. Final simplification 59.4%

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

Alternative 22: 56.6% accurate, 17.0× speedup

Math

\[\begin{array}{l} cosTheta_O\_m = \left|cosTheta\_O\right| \\ cosTheta_O\_s = \mathsf{copysign}\left(1, cosTheta\_O\right) \\ [cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])\\ \\ cosTheta\_O\_s \cdot \left(v \cdot \left(\left(cosTheta\_O\_m \cdot cosTheta\_i\right) \cdot 3\right)\right) \end{array} \]

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (* cosTheta_O_s (* v (* (* cosTheta_O_m cosTheta_i) 3.0))))

C

cosTheta_O\_m = fabs(cosTheta_O);
cosTheta_O\_s = copysign(1.0, cosTheta_O);
assert(cosTheta_i < cosTheta_O_m && cosTheta_O_m < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_O_s, float cosTheta_i, float cosTheta_O_m, float sinTheta_i, float sinTheta_O, float v) {
	return cosTheta_O_s * (v * ((cosTheta_O_m * cosTheta_i) * 3.0f));
}

Fortran

cosTheta_O\_m = abs(costheta_o)
cosTheta_O\_s = copysign(1.0d0, costheta_o)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_o_s, costheta_i, costheta_o_m, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_o_s
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o_m
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = costheta_o_s * (v * ((costheta_o_m * costheta_i) * 3.0e0))
end function

Julia

cosTheta_O\_m = abs(cosTheta_O)
cosTheta_O\_s = copysign(1.0, cosTheta_O)
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])
function code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	return Float32(cosTheta_O_s * Float32(v * Float32(Float32(cosTheta_O_m * cosTheta_i) * Float32(3.0))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_O_s, cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, v)
	tmp = cosTheta_O_s * (v * ((cosTheta_O_m * cosTheta_i) * single(3.0)));
end

Derivation

1. Initial program 98.6%

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

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

   8. lift-sinh.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. times-frac N/A

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

4. Applied egg-rr 98.6%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. lower--.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. rec-exp N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 98.4

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

7. Simplified 98.4%

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

8. Taylor expanded in v around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 65.0

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

10. Simplified 65.0%

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

11. Taylor expanded in v around 0

    \[\leadsto \color{blue}{3 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_i \cdot v\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto 3 \cdot \color{blue}{\left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot v\right)} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{\left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right) \cdot v} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{v \cdot \left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]

    4. lower-*.f32 N/A

       \[\leadsto \color{blue}{v \cdot \left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]

    5. lower-*.f32 N/A

       \[\leadsto v \cdot \color{blue}{\left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]

    6. lower-*.f32 57.9

       \[\leadsto v \cdot \left(3 \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right) \]

13. Simplified 57.9%

    \[\leadsto \color{blue}{v \cdot \left(3 \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)} \]

14. Final simplification 57.9%

    \[\leadsto v \cdot \left(\left(cosTheta\_O \cdot cosTheta\_i\right) \cdot 3\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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)))