HairBSDF, Mp, upper

Percentage Accurate: 98.6% → 98.5%

Time: 16.5s

Alternatives: 8

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 1.8× speedup

Math

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

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
  (*
   (* 2.0 (* (/ cosTheta_i (sinh (/ 1.0 v))) (/ (/ 1.0 v) 2.0)))
   (/ cosTheta_O_m (* 2.0 v)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r* 98.8%

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

   2. associate-*r* 98.8%

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

   3. times-frac 98.8%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.7%

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

   2. associate-/r* 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in sinTheta_i around 0 98.5%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.5%

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

    3. distribute-neg-frac 98.5%

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

    4. metadata-eval 98.5%

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

11. Simplified 98.5%

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

    1. clear-num 98.4%

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

    2. inv-pow 98.4%

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

    3. div-inv 98.4%

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

    4. div-inv 98.4%

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

    5. neg-mul-1 98.4%

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

    6. sinh-undef 98.4%

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

    7. clear-num 98.4%

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

13. Applied egg-rr 98.4%

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

    1. *-commutative 98.4%

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

    2. unpow-prod-down 98.5%

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

    3. inv-pow 98.5%

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

    4. clear-num 98.5%

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

    5. inv-pow 98.5%

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

    6. div-inv 98.5%

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

    7. div-inv 98.7%

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

    8. *-commutative 98.7%

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

    9. times-frac 98.7%

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

15. Applied egg-rr 98.7%

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

16. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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 (* 2.0 v))
   (* 2.0 (* 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 / (2.0f * v)) * (2.0f * (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 / (2.0e0 * v)) * (2.0e0 * (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(cosTheta_O_m / Float32(Float32(2.0) * v)) * Float32(Float32(2.0) * Float32(cosTheta_i * Float32(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(2.0) * v)) * (single(2.0) * (cosTheta_i * ((single(0.5) / v) / sinh((single(1.0) / v))))));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r* 98.8%

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

   2. associate-*r* 98.8%

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

   3. times-frac 98.8%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.7%

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

   2. associate-/r* 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in sinTheta_i around 0 98.5%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.5%

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

    3. distribute-neg-frac 98.5%

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

    4. metadata-eval 98.5%

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

11. Simplified 98.5%

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

    1. *-un-lft-identity 98.5%

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

    2. associate-/l/ 98.6%

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

    3. *-commutative 98.6%

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

    4. div-inv 98.6%

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

    5. neg-mul-1 98.6%

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

    6. sinh-undef 98.6%

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

13. Applied egg-rr 98.6%

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

    1. *-commutative 98.6%

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

    2. associate-*l/ 98.6%

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

    3. times-frac 98.5%

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

    4. *-rgt-identity 98.5%

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

    5. associate-*r/ 98.6%

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

    6. associate-*r* 98.7%

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

    7. associate-/r* 98.7%

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

    8. metadata-eval 98.7%

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

    9. associate-*r/ 98.7%

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

    10. associate-*l/ 98.7%

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

    11. metadata-eval 98.7%

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

15. Simplified 98.7%

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

16. Final simplification 98.7%

    \[\leadsto \frac{cosTheta\_O}{2 \cdot v} \cdot \left(2 \cdot \left(cosTheta\_i \cdot \frac{\frac{0.5}{v}}{\sinh \left(\frac{1}{v}\right)}\right)\right) \]
17. Add Preprocessing

Alternative 3: 98.3% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
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 v) (/ (/ cosTheta_i 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 * (0.5f * ((cosTheta_O_m / v) * ((cosTheta_i / 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 * (0.5e0 * ((costheta_o_m / v) * ((costheta_i / 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(0.5) * Float32(Float32(cosTheta_O_m / v) * Float32(Float32(cosTheta_i / 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 * (single(0.5) * ((cosTheta_O_m / v) * ((cosTheta_i / v) / sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r* 98.8%

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

   2. associate-*r* 98.8%

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

   3. times-frac 98.8%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.7%

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

   2. associate-/r* 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in sinTheta_i around 0 98.5%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.5%

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

    3. distribute-neg-frac 98.5%

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

    4. metadata-eval 98.5%

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

11. Simplified 98.5%

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

    1. clear-num 98.4%

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

    2. inv-pow 98.4%

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

    3. div-inv 98.4%

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

    4. div-inv 98.4%

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

    5. neg-mul-1 98.4%

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

    6. sinh-undef 98.4%

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

    7. clear-num 98.4%

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

13. Applied egg-rr 98.4%

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

    1. associate-*r/ 98.2%

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

15. Applied egg-rr 98.2%

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

    1. *-commutative 98.2%

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

    2. times-frac 98.5%

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

    3. *-lft-identity 98.5%

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

    4. associate-/r* 98.5%

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

    5. associate-/r* 98.6%

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

    6. *-rgt-identity 98.6%

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

    7. associate-*r* 98.6%

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

    8. *-commutative 98.6%

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

    9. times-frac 98.6%

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

    10. associate-/l/ 98.5%

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

    11. *-lft-identity 98.5%

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

    12. metadata-eval 98.5%

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

    13. associate-*r* 98.5%

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

    14. *-commutative 98.5%

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

    15. associate-*l* 98.3%

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

    16. *-lft-identity 98.3%

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

17. Simplified 98.3%

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

18. Final simplification 98.3%

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

Alternative 4: 98.4% accurate, 1.9× speedup

Math

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

FPCore

cosTheta_O\_m = (fabs.f32 cosTheta_O)
cosTheta_O\_s = (copysign.f32 #s(literal 1 binary32) cosTheta_O)
NOTE: cosTheta_i, cosTheta_O_m, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_O_s cosTheta_i cosTheta_O_m sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  cosTheta_O_s
  (* (/ cosTheta_O_m (* 2.0 v)) (/ cosTheta_i (* 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 / (2.0f * v)) * (cosTheta_i / (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 / (2.0e0 * v)) * (costheta_i / (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(cosTheta_O_m / Float32(Float32(2.0) * v)) * Float32(cosTheta_i / Float32(v * sinh(Float32(Float32(1.0) / v))))))
end

MATLAB

cosTheta_O\_m = abs(cosTheta_O);
cosTheta_O\_s = sign(cosTheta_O) * abs(1.0);
cosTheta_i, 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(2.0) * v)) * (cosTheta_i / (v * sinh((single(1.0) / v)))));
end

Derivation

1. Initial program 98.7%

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

   1. associate-*r/ 98.7%

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

   2. associate-/l/ 98.6%

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

   3. remove-double-neg 98.6%

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

   4. distribute-rgt-neg-out 98.6%

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

   5. distribute-rgt-neg-out 98.6%

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

   6. distribute-lft-neg-in 98.6%

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

   7. associate-*r/ 98.6%

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

   8. associate-/l/ 98.7%

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

   9. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. associate-*r* 98.8%

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

   2. associate-*r* 98.8%

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

   3. times-frac 98.8%

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

6. Applied egg-rr 98.5%

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

   1. times-frac 98.7%

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

   2. associate-/r* 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in sinTheta_i around 0 98.5%

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

    1. associate-/r* 98.5%

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

    2. rec-exp 98.5%

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

    3. distribute-neg-frac 98.5%

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

    4. metadata-eval 98.5%

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

11. Simplified 98.5%

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

    1. clear-num 98.4%

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

    2. inv-pow 98.4%

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

    3. div-inv 98.4%

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

    4. div-inv 98.4%

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

    5. neg-mul-1 98.4%

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

    6. sinh-undef 98.4%

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

    7. clear-num 98.4%

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

13. Applied egg-rr 98.4%

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

    1. associate-*r/ 98.2%

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

15. Applied egg-rr 98.2%

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

    1. associate-/l* 98.5%

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

    2. *-lft-identity 98.5%

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

    3. associate-/l/ 98.6%

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

    4. *-commutative 98.6%

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

17. Simplified 98.6%

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

18. Final simplification 98.6%

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

Alternative 5: 58.7% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 62.1%

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

6. Taylor expanded in v around inf 62.1%

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

   1. associate-/l* 62.1%

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

8. Simplified 62.1%

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

   1. associate-*r/ 62.1%

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

   2. clear-num 62.3%

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

10. Applied egg-rr 62.3%

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

11. Final simplification 62.3%

    \[\leadsto 0.5 \cdot \frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \]
12. Add Preprocessing

Alternative 6: 58.3% accurate, 31.4× speedup

Math

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

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 (* 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(cosTheta_O_m * Float32(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 98.7%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 62.1%

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

6. Taylor expanded in v around inf 62.1%

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

   1. associate-/l* 62.1%

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

8. Simplified 62.1%

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

9. Final simplification 62.1%

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

Alternative 7: 58.3% accurate, 31.4× speedup

Math

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

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_i cosTheta_O_m) 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_i * cosTheta_O_m) / 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_i * costheta_o_m) / 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_i * cosTheta_O_m) / 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_i * cosTheta_O_m) / v));
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 62.1%

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

6. Taylor expanded in v around inf 62.1%

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

7. Final simplification 62.1%

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

Alternative 8: 58.3% accurate, 31.4× speedup

Math

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

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_i cosTheta_O_m)) 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_i * cosTheta_O_m)) / 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_i * costheta_o_m)) / 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_i * cosTheta_O_m)) / 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_i * cosTheta_O_m)) / v);
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.5%

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

5. Taylor expanded in v around inf 62.1%

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

6. Taylor expanded in v around inf 62.1%

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

   1. associate-/l* 62.1%

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

8. Simplified 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-*r/ 62.1%

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

   3. associate-*l/ 62.1%

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

10. Applied egg-rr 62.1%

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

11. Final simplification 62.1%

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

Reproduce

herbie shell --seed 2024085 
(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)))