HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.5%

Time: 16.6s

Alternatives: 9

Speedup: N/A×

Specification

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

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf((1.0f / (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(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) - Float32(Float32(sinTheta_i * sinTheta_O) / v)) - Float32(Float32(1.0) / v)) + Float32(0.6931)) + log(Float32(Float32(1.0) / Float32(Float32(2.0) * v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (single(1.0) / v)) + single(0.6931)) + log((single(1.0) / (single(2.0) * v)))));
end

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := \log \left(\frac{0.5}{v}\right)\\ {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v} + \left(0.6931 + t\_0\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(\frac{0.6931 + \left(t\_0 + \left(\frac{-1}{v} - sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}{cosTheta\_O} + \frac{cosTheta\_i}{v}\right)}{2}\right)} \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (log (/ 0.5 v))))
   (*
    (pow
     E
     (/
      (+
       (/ (+ (fma cosTheta_i cosTheta_O (* sinTheta_i (- sinTheta_O))) -1.0) v)
       (+ 0.6931 t_0))
      2.0))
    (pow
     E
     (/
      (*
       cosTheta_O
       (+
        (/
         (+ 0.6931 (+ t_0 (- (/ -1.0 v) (* sinTheta_O (/ sinTheta_i v)))))
         cosTheta_O)
        (/ cosTheta_i v)))
      2.0)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = logf((0.5f / v));
	return powf(((float) M_E), ((((fmaf(cosTheta_i, cosTheta_O, (sinTheta_i * -sinTheta_O)) + -1.0f) / v) + (0.6931f + t_0)) / 2.0f)) * powf(((float) M_E), ((cosTheta_O * (((0.6931f + (t_0 + ((-1.0f / v) - (sinTheta_O * (sinTheta_i / v))))) / cosTheta_O) + (cosTheta_i / v))) / 2.0f));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = log(Float32(Float32(0.5) / v))
	return Float32((Float32(exp(1)) ^ Float32(Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(sinTheta_i * Float32(-sinTheta_O))) + Float32(-1.0)) / v) + Float32(Float32(0.6931) + t_0)) / Float32(2.0))) * (Float32(exp(1)) ^ Float32(Float32(cosTheta_O * Float32(Float32(Float32(Float32(0.6931) + Float32(t_0 + Float32(Float32(Float32(-1.0) / v) - Float32(sinTheta_O * Float32(sinTheta_i / v))))) / cosTheta_O) + Float32(cosTheta_i / v))) / Float32(2.0))))
end

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.5%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.5%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.5%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   2. exp-prod 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   3. associate--r+ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)}} \]

   5. associate-*r/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)} \]

   6. associate-*r/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)} \]

   7. sub-div 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)} \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)}} \]
7. Step-by-step derivation

   1. sqr-pow 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)}} \]

   2. exp-1-e 99.6%

      \[\leadsto {\color{blue}{e}}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   3. associate--r- 99.6%

      \[\leadsto {e}^{\left(\frac{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   4. sub-div 99.6%

      \[\leadsto {e}^{\left(\frac{\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   5. fma-neg 99.6%

      \[\leadsto {e}^{\left(\frac{\frac{\color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right)} - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   6. exp-1-e 99.6%

      \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {\color{blue}{e}}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}} \]

9. Taylor expanded in cosTheta_O around -inf 99.6%

   \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\color{blue}{-1 \cdot \left(cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + -1 \cdot \frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_O}\right)\right)}}{2}\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\color{blue}{-cosTheta\_O \cdot \left(-1 \cdot \frac{cosTheta\_i}{v} + -1 \cdot \frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_O}\right)}}{2}\right)} \]

    2. distribute-rgt-neg-in 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\color{blue}{cosTheta\_O \cdot \left(-\left(-1 \cdot \frac{cosTheta\_i}{v} + -1 \cdot \frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_O}\right)\right)}}{2}\right)} \]

    3. mul-1-neg 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(-\left(-1 \cdot \frac{cosTheta\_i}{v} + \color{blue}{\left(-\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_O}\right)}\right)\right)}{2}\right)} \]

    4. unsub-neg 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(-\color{blue}{\left(-1 \cdot \frac{cosTheta\_i}{v} - \frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_O}\right)}\right)}{2}\right)} \]

    5. mul-1-neg 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(-\left(\color{blue}{\left(-\frac{cosTheta\_i}{v}\right)} - \frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_O}\right)\right)}{2}\right)} \]

    6. distribute-neg-frac2 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(-\left(\color{blue}{\frac{cosTheta\_i}{-v}} - \frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}{cosTheta\_O}\right)\right)}{2}\right)} \]

    7. associate--l+ 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(-\left(\frac{cosTheta\_i}{-v} - \frac{\color{blue}{0.6931 + \left(\log \left(\frac{0.5}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}}{cosTheta\_O}\right)\right)}{2}\right)} \]

    8. associate-/l* 99.6%

       \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(-\left(\frac{cosTheta\_i}{-v} - \frac{0.6931 + \left(\log \left(\frac{0.5}{v}\right) - \left(\frac{1}{v} + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\right)\right)}{cosTheta\_O}\right)\right)}{2}\right)} \]

11. Simplified 99.6%

    \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\color{blue}{cosTheta\_O \cdot \left(-\left(\frac{cosTheta\_i}{-v} - \frac{0.6931 + \left(\log \left(\frac{0.5}{v}\right) - \left(\frac{1}{v} + sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}{cosTheta\_O}\right)\right)}}{2}\right)} \]

12. Final simplification 99.6%

    \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{cosTheta\_O \cdot \left(\frac{0.6931 + \left(\log \left(\frac{0.5}{v}\right) + \left(\frac{-1}{v} - sinTheta\_O \cdot \frac{sinTheta\_i}{v}\right)\right)}{cosTheta\_O} + \frac{cosTheta\_i}{v}\right)}{2}\right)} \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := 0.6931 + \log \left(\frac{0.5}{v}\right)\\ {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v} + t\_0}{2}\right)} \cdot {e}^{\left(\frac{t\_0 + \frac{cosTheta\_i \cdot cosTheta\_O + -1}{v}}{2}\right)} \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (+ 0.6931 (log (/ 0.5 v)))))
   (*
    (pow
     E
     (/
      (+
       (/ (+ (fma cosTheta_i cosTheta_O (* sinTheta_i (- sinTheta_O))) -1.0) v)
       t_0)
      2.0))
    (pow E (/ (+ t_0 (/ (+ (* cosTheta_i cosTheta_O) -1.0) v)) 2.0)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = 0.6931f + logf((0.5f / v));
	return powf(((float) M_E), ((((fmaf(cosTheta_i, cosTheta_O, (sinTheta_i * -sinTheta_O)) + -1.0f) / v) + t_0) / 2.0f)) * powf(((float) M_E), ((t_0 + (((cosTheta_i * cosTheta_O) + -1.0f) / v)) / 2.0f));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(0.6931) + log(Float32(Float32(0.5) / v)))
	return Float32((Float32(exp(1)) ^ Float32(Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(sinTheta_i * Float32(-sinTheta_O))) + Float32(-1.0)) / v) + t_0) / Float32(2.0))) * (Float32(exp(1)) ^ Float32(Float32(t_0 + Float32(Float32(Float32(cosTheta_i * cosTheta_O) + Float32(-1.0)) / v)) / Float32(2.0))))
end

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.5%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.5%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.5%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   2. exp-prod 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   3. associate--r+ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)}} \]

   5. associate-*r/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)} \]

   6. associate-*r/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)} \]

   7. sub-div 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)} \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)\right)}} \]
7. Step-by-step derivation

   1. sqr-pow 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)}} \]

   2. exp-1-e 99.6%

      \[\leadsto {\color{blue}{e}}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   3. associate--r- 99.6%

      \[\leadsto {e}^{\left(\frac{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   4. sub-div 99.6%

      \[\leadsto {e}^{\left(\frac{\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   5. fma-neg 99.6%

      \[\leadsto {e}^{\left(\frac{\frac{\color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right)} - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

   6. exp-1-e 99.6%

      \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {\color{blue}{e}}^{\left(\frac{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \left(\frac{1}{v} - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)} \]

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}} \]

9. Taylor expanded in cosTheta_i around inf 99.6%

   \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i} - 1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \]

10. Final simplification 99.6%

    \[\leadsto {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v} + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \frac{cosTheta\_i \cdot cosTheta\_O + -1}{v}}{2}\right)} \]
11. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} t_0 := {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v} + 0.6931}{2}\right)}\\ \frac{0.5}{v} \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (pow
          E
          (/
           (+
            (/
             (+ (fma cosTheta_i cosTheta_O (* sinTheta_i (- sinTheta_O))) -1.0)
             v)
            0.6931)
           2.0))))
   (* (/ 0.5 v) (* t_0 t_0))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = powf(((float) M_E), ((((fmaf(cosTheta_i, cosTheta_O, (sinTheta_i * -sinTheta_O)) + -1.0f) / v) + 0.6931f) / 2.0f));
	return (0.5f / v) * (t_0 * t_0);
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(exp(1)) ^ Float32(Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(sinTheta_i * Float32(-sinTheta_O))) + Float32(-1.0)) / v) + Float32(0.6931)) / Float32(2.0))
	return Float32(Float32(Float32(0.5) / v) * Float32(t_0 * t_0))
end

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.5%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.5%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{1 \cdot \left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931\right)}} \]

   2. exp-prod 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931\right)}} \]

   3. fma-define 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)}\right) + 0.6931\right)} \]

   4. associate--r+ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + 0.6931\right)} \]

   5. associate-*r/ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right)} \]

   6. associate-*r/ 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) - \frac{1}{v}\right) + 0.6931\right)} \]

   7. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + 0.6931\right)} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931\right)}} \]
7. Step-by-step derivation

   1. sqr-pow 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left({\left(e^{1}\right)}^{\left(\frac{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}{2}\right)}\right)} \]

   2. exp-1-e 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left({\color{blue}{e}}^{\left(\frac{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}{2}\right)}\right) \]

   3. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + 0.6931}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}{2}\right)}\right) \]

   4. fma-neg 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\frac{\color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right)} - 1}{v} + 0.6931}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}{2}\right)}\right) \]

   5. exp-1-e 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}{2}\right)} \cdot {\color{blue}{e}}^{\left(\frac{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}{2}\right)}\right) \]

   6. sub-div 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}{2}\right)} \cdot {e}^{\left(\frac{\color{blue}{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}} + 0.6931}{2}\right)}\right) \]

   7. fma-neg 99.6%

      \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}{2}\right)} \cdot {e}^{\left(\frac{\frac{\color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right)} - 1}{v} + 0.6931}{2}\right)}\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left({e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}{2}\right)} \cdot {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}{2}\right)}\right)} \]

9. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v} + 0.6931}{2}\right)} \cdot {e}^{\left(\frac{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(-sinTheta\_O\right)\right) + -1}{v} + 0.6931}{2}\right)}\right) \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{v} \cdot {e}^{\left(0.6931 + \frac{-1}{v}\right)} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (pow E (+ 0.6931 (/ -1.0 v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(((float) M_E), (0.6931f + (-1.0f / v)));
}

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (Float32(exp(1)) ^ Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(2.71828182845904523536) ^ (single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.5%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.5%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.5%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]

6. Taylor expanded in cosTheta_i around -inf 99.5%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{-1 \cdot \left(cosTheta\_i \cdot \left(-1 \cdot \frac{cosTheta\_O}{v} + -1 \cdot \frac{0.6931 - \frac{1}{v}}{cosTheta\_i}\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*r* 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-1 \cdot cosTheta\_i\right) \cdot \left(-1 \cdot \frac{cosTheta\_O}{v} + -1 \cdot \frac{0.6931 - \frac{1}{v}}{cosTheta\_i}\right)}} \]

   2. mul-1-neg 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-cosTheta\_i\right)} \cdot \left(-1 \cdot \frac{cosTheta\_O}{v} + -1 \cdot \frac{0.6931 - \frac{1}{v}}{cosTheta\_i}\right)} \]

   3. mul-1-neg 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-cosTheta\_i\right) \cdot \left(-1 \cdot \frac{cosTheta\_O}{v} + \color{blue}{\left(-\frac{0.6931 - \frac{1}{v}}{cosTheta\_i}\right)}\right)} \]

   4. unsub-neg 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-cosTheta\_i\right) \cdot \color{blue}{\left(-1 \cdot \frac{cosTheta\_O}{v} - \frac{0.6931 - \frac{1}{v}}{cosTheta\_i}\right)}} \]

   5. associate-*r/ 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-cosTheta\_i\right) \cdot \left(\color{blue}{\frac{-1 \cdot cosTheta\_O}{v}} - \frac{0.6931 - \frac{1}{v}}{cosTheta\_i}\right)} \]

   6. mul-1-neg 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-cosTheta\_i\right) \cdot \left(\frac{\color{blue}{-cosTheta\_O}}{v} - \frac{0.6931 - \frac{1}{v}}{cosTheta\_i}\right)} \]

   7. sub-neg 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-cosTheta\_i\right) \cdot \left(\frac{-cosTheta\_O}{v} - \frac{\color{blue}{0.6931 + \left(-\frac{1}{v}\right)}}{cosTheta\_i}\right)} \]

   8. distribute-neg-frac 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-cosTheta\_i\right) \cdot \left(\frac{-cosTheta\_O}{v} - \frac{0.6931 + \color{blue}{\frac{-1}{v}}}{cosTheta\_i}\right)} \]

   9. metadata-eval 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(-cosTheta\_i\right) \cdot \left(\frac{-cosTheta\_O}{v} - \frac{0.6931 + \frac{\color{blue}{-1}}{v}}{cosTheta\_i}\right)} \]

8. Simplified 99.5%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(-cosTheta\_i\right) \cdot \left(\frac{-cosTheta\_O}{v} - \frac{0.6931 + \frac{-1}{v}}{cosTheta\_i}\right)}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{1 \cdot \left(\left(-cosTheta\_i\right) \cdot \left(\frac{-cosTheta\_O}{v} - \frac{0.6931 + \frac{-1}{v}}{cosTheta\_i}\right)\right)}} \]

   2. exp-prod 99.5%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(-cosTheta\_i\right) \cdot \left(\frac{-cosTheta\_O}{v} - \frac{0.6931 + \frac{-1}{v}}{cosTheta\_i}\right)\right)}} \]

   3. e-exp-1 99.5%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{e}}^{\left(\left(-cosTheta\_i\right) \cdot \left(\frac{-cosTheta\_O}{v} - \frac{0.6931 + \frac{-1}{v}}{cosTheta\_i}\right)\right)} \]

   4. distribute-frac-neg 99.5%

      \[\leadsto \frac{0.5}{v} \cdot {e}^{\left(\left(-cosTheta\_i\right) \cdot \left(\color{blue}{\left(-\frac{cosTheta\_O}{v}\right)} - \frac{0.6931 + \frac{-1}{v}}{cosTheta\_i}\right)\right)} \]

10. Applied egg-rr 99.5%

    \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{e}^{\left(\left(-cosTheta\_i\right) \cdot \left(\left(-\frac{cosTheta\_O}{v}\right) - \frac{0.6931 + \frac{-1}{v}}{cosTheta\_i}\right)\right)}} \]

12. Simplified 99.5%

    \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{e}^{\left(cosTheta\_i \cdot \left(-\left(\frac{\frac{1}{v} + -0.6931}{cosTheta\_i} - \frac{cosTheta\_O}{v}\right)\right)\right)}} \]

13. Taylor expanded in cosTheta_i around 0 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {e}^{\color{blue}{\left(0.6931 - \frac{1}{v}\right)}} \]

14. Final simplification 99.6%

    \[\leadsto \frac{0.5}{v} \cdot {e}^{\left(0.6931 + \frac{-1}{v}\right)} \]
15. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ -1.0 v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * expf((0.6931f + (-1.0f / v)));
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 + ((-1.0e0) / v)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * exp(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.5%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2 \cdot v}\right)} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]

   3. rem-exp-log 99.5%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   4. associate-/r* 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   5. metadata-eval 99.5%

      \[\leadsto \frac{\color{blue}{0.5}}{v} \cdot e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

   6. associate--l- 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + 0.6931} \]

   7. associate-/l* 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931} \]

   8. associate-/l* 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + 0.6931} \]

   9. fma-define 99.5%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \color{blue}{\mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}\right) + 0.6931} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)\right) + 0.6931}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around 0 99.5%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}} \]

6. Taylor expanded in cosTheta_O around 0 99.5%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{e^{0.6931 - \frac{1}{v}}} \]

7. Final simplification 99.5%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \]
8. Add Preprocessing

Alternative 6: 13.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((sintheta_i * -sintheta_o) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.5%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.5%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.5%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in sinTheta_i around inf 14.6%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}} \]
6. Step-by-step derivation

   1. associate-*r/ 14.6%

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

   2. mul-1-neg 14.6%

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

   3. distribute-lft-neg-out 14.6%

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

   4. *-commutative 14.6%

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

7. Simplified 14.6%

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

Alternative 7: 13.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((costheta_i * (costheta_o / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.5%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.5%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.5%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around inf 79.7%

   \[\leadsto e^{\color{blue}{cosTheta\_i \cdot \left(\left(0.6931 \cdot \frac{1}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\color{blue}{\frac{0.6931 \cdot 1}{cosTheta\_i}} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   2. metadata-eval 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{\color{blue}{0.6931}}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   3. *-commutative 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{\color{blue}{v \cdot cosTheta\_i}} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   4. associate-/l* 83.6%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{cosTheta\_i \cdot v}}\right)\right)} \]

   5. *-commutative 83.6%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{v \cdot cosTheta\_i}}\right)\right)} \]

7. Simplified 83.6%

   \[\leadsto e^{\color{blue}{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + sinTheta\_O \cdot \frac{sinTheta\_i}{v \cdot cosTheta\_i}\right)\right)}} \]

8. Taylor expanded in cosTheta_i around inf 11.3%

   \[\leadsto e^{cosTheta\_i \cdot \color{blue}{\frac{cosTheta\_O}{v}}} \]
9. Add Preprocessing

Alternative 8: 13.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp((costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.5%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.5%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.5%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around inf 79.7%

   \[\leadsto e^{\color{blue}{cosTheta\_i \cdot \left(\left(0.6931 \cdot \frac{1}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\color{blue}{\frac{0.6931 \cdot 1}{cosTheta\_i}} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   2. metadata-eval 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{\color{blue}{0.6931}}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   3. *-commutative 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{\color{blue}{v \cdot cosTheta\_i}} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   4. associate-/l* 83.6%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{cosTheta\_i \cdot v}}\right)\right)} \]

   5. *-commutative 83.6%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{v \cdot cosTheta\_i}}\right)\right)} \]

7. Simplified 83.6%

   \[\leadsto e^{\color{blue}{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + sinTheta\_O \cdot \frac{sinTheta\_i}{v \cdot cosTheta\_i}\right)\right)}} \]

8. Taylor expanded in cosTheta_i around inf 11.3%

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

9. Taylor expanded in cosTheta_i around inf 11.3%

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

    1. associate-/l* 11.3%

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

11. Simplified 11.3%

    \[\leadsto \color{blue}{e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
12. Add Preprocessing

Alternative 9: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.5%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. associate--l- 99.5%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-/l* 99.5%

      \[\leadsto e^{\left(\color{blue}{cosTheta\_i \cdot \frac{cosTheta\_O}{v}} - \left(\frac{sinTheta\_i \cdot sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-/l* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(\color{blue}{sinTheta\_i \cdot \frac{sinTheta\_O}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. associate-/r* 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{e^{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in cosTheta_i around inf 79.7%

   \[\leadsto e^{\color{blue}{cosTheta\_i \cdot \left(\left(0.6931 \cdot \frac{1}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\color{blue}{\frac{0.6931 \cdot 1}{cosTheta\_i}} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   2. metadata-eval 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{\color{blue}{0.6931}}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{cosTheta\_i \cdot v} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   3. *-commutative 79.7%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{\color{blue}{v \cdot cosTheta\_i}} + \frac{sinTheta\_O \cdot sinTheta\_i}{cosTheta\_i \cdot v}\right)\right)} \]

   4. associate-/l* 83.6%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + \color{blue}{sinTheta\_O \cdot \frac{sinTheta\_i}{cosTheta\_i \cdot v}}\right)\right)} \]

   5. *-commutative 83.6%

      \[\leadsto e^{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + sinTheta\_O \cdot \frac{sinTheta\_i}{\color{blue}{v \cdot cosTheta\_i}}\right)\right)} \]

7. Simplified 83.6%

   \[\leadsto e^{\color{blue}{cosTheta\_i \cdot \left(\left(\frac{0.6931}{cosTheta\_i} + \left(\frac{cosTheta\_O}{v} + \frac{\log \left(\frac{0.5}{v}\right)}{cosTheta\_i}\right)\right) - \left(\frac{1}{v \cdot cosTheta\_i} + sinTheta\_O \cdot \frac{sinTheta\_i}{v \cdot cosTheta\_i}\right)\right)}} \]

8. Taylor expanded in cosTheta_i around inf 11.3%

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

9. Taylor expanded in cosTheta_i around 0 6.5%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (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))) (and (<= -1.5707964 v) (<= v 0.1)))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))