HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 12.6s

Alternatives: 10

Speedup: 1.0×

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: 99.6% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{{\left({e}^{2}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}\right)}^{2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  (sqrt
   (pow
    (pow E 2.0)
    (/
     (+
      (-
       (* cosTheta_O (/ cosTheta_i v))
       (fma sinTheta_O (/ sinTheta_i v) (/ 1.0 v)))
      (+ 0.6931 (log (/ 0.5 v))))
     2.0)))
  2.0))

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 99.6%

      \[\leadsto \color{blue}{\sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot \sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}} \]

   2. pow2 99.6%

      \[\leadsto \color{blue}{{\left(\sqrt{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2}} \]

   3. associate-/r/ 99.6%

      \[\leadsto {\left(\sqrt{e^{\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2} \]

   4. *-commutative 99.6%

      \[\leadsto {\left(\sqrt{e^{\left(\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2} \]

   5. associate-*l/ 99.6%

      \[\leadsto {\left(\sqrt{e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2} \]

   6. *-commutative 99.6%

      \[\leadsto {\left(\sqrt{e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2} \]

   7. fma-def 99.6%

      \[\leadsto {\left(\sqrt{e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(\sqrt{e^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}\right)}^{2}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto {\left(\sqrt{e^{\color{blue}{1 \cdot \left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}}}\right)}^{2} \]

   2. pow-exp 99.6%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}}}\right)}^{2} \]

   3. sqr-pow 99.6%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}}\right)}^{2} \]

   4. pow-prod-down 99.6%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left(e^{1} \cdot e^{1}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}}\right)}^{2} \]

   5. pow2 99.6%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left({\left(e^{1}\right)}^{2}\right)}}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}\right)}^{2} \]

   6. exp-1-e 99.6%

      \[\leadsto {\left(\sqrt{{\left({\color{blue}{e}}^{2}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}\right)}^{2} \]

   7. associate-+l- 99.6%

      \[\leadsto {\left(\sqrt{{\left({e}^{2}\right)}^{\left(\frac{\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right) - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}}{2}\right)}}\right)}^{2} \]

   8. inv-pow 99.6%

      \[\leadsto {\left(\sqrt{{\left({e}^{2}\right)}^{\left(\frac{cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \color{blue}{{v}^{-1}}\right) - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)}}\right)}^{2} \]

7. Applied egg-rr 99.6%

   \[\leadsto {\left(\sqrt{\color{blue}{{\left({e}^{2}\right)}^{\left(\frac{cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, {v}^{-1}\right) - \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}{2}\right)}}}\right)}^{2} \]
8. Step-by-step derivation

   1. associate--r- 99.6%

      \[\leadsto {\left(\sqrt{{\left({e}^{2}\right)}^{\left(\frac{\color{blue}{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, {v}^{-1}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}}{2}\right)}}\right)}^{2} \]

   2. unpow-1 99.6%

      \[\leadsto {\left(\sqrt{{\left({e}^{2}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \color{blue}{\frac{1}{v}}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}\right)}^{2} \]

9. Simplified 99.6%

   \[\leadsto {\left(\sqrt{\color{blue}{{\left({e}^{2}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}}\right)}^{2} \]

10. Final simplification 99.6%

    \[\leadsto {\left(\sqrt{{\left({e}^{2}\right)}^{\left(\frac{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}{2}\right)}}\right)}^{2} \]

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  E
  (+
   (-
    (* cosTheta_O (/ cosTheta_i v))
    (fma sinTheta_O (/ sinTheta_i v) (/ 1.0 v)))
   (+ 0.6931 (log (/ 0.5 v))))))

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   2. exp-prod 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   3. associate-/r/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   4. *-commutative 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   5. associate-*l/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   6. *-commutative 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   7. fma-def 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

6. Final simplification 99.6%

   \[\leadsto {e}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

Alternative 3: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* (/ 0.5 v) (exp 0.6931))
  (exp
   (-
    (* cosTheta_O (/ cosTheta_i v))
    (fma sinTheta_O (/ sinTheta_i v) (pow v -1.0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((0.5f / v) * expf(0.6931f)) * expf(((cosTheta_O * (cosTheta_i / v)) - fmaf(sinTheta_O, (sinTheta_i / v), powf(v, -1.0f))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(0.5) / v) * exp(Float32(0.6931))) * exp(Float32(Float32(cosTheta_O * Float32(cosTheta_i / v)) - fma(sinTheta_O, Float32(sinTheta_i / v), (v ^ Float32(-1.0))))))
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto e^{\color{blue}{1 \cdot \left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   2. exp-prod 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]

   3. associate-/r/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{\frac{cosTheta_i}{v} \cdot cosTheta_O} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   4. *-commutative 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   5. associate-*l/ 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\color{blue}{\frac{sinTheta_i}{v} \cdot sinTheta_O} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   6. *-commutative 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \left(\color{blue}{sinTheta_O \cdot \frac{sinTheta_i}{v}} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

   7. fma-def 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \color{blue}{\mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)}\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative 99.6%

      \[\leadsto {\left(e^{1}\right)}^{\color{blue}{\left(\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)\right)}} \]

   2. unpow-prod-up 99.6%

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)} \cdot {\left(e^{1}\right)}^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)}} \]

   3. pow-exp 99.6%

      \[\leadsto \color{blue}{e^{1 \cdot \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \cdot {\left(e^{1}\right)}^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \]

   4. *-un-lft-identity 99.6%

      \[\leadsto e^{\color{blue}{0.6931 + \log \left(\frac{0.5}{v}\right)}} \cdot {\left(e^{1}\right)}^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \]

   5. add-log-exp 99.6%

      \[\leadsto e^{\color{blue}{\log \left(e^{0.6931}\right)} + \log \left(\frac{0.5}{v}\right)} \cdot {\left(e^{1}\right)}^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \]

   6. log-prod 99.6%

      \[\leadsto e^{\color{blue}{\log \left(e^{0.6931} \cdot \frac{0.5}{v}\right)}} \cdot {\left(e^{1}\right)}^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \]

   7. add-exp-log 99.6%

      \[\leadsto \color{blue}{\left(e^{0.6931} \cdot \frac{0.5}{v}\right)} \cdot {\left(e^{1}\right)}^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \]

   8. *-commutative 99.6%

      \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right)} \cdot {\left(e^{1}\right)}^{\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)} \]

   9. pow-exp 99.6%

      \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot \color{blue}{e^{1 \cdot \left(cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)\right)}} \]

   10. *-un-lft-identity 99.6%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot e^{\color{blue}{cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \frac{1}{v}\right)}} \]

   11. inv-pow 99.6%

       \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot e^{cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, \color{blue}{{v}^{-1}}\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot e^{cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, {v}^{-1}\right)}} \]

8. Final simplification 99.6%

   \[\leadsto \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \cdot e^{cosTheta_O \cdot \frac{cosTheta_i}{v} - \mathsf{fma}\left(sinTheta_O, \frac{sinTheta_i}{v}, {v}^{-1}\right)} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+ 0.6931 (log (/ 0.5 v)))
   (+
    (/ cosTheta_i (/ v cosTheta_O))
    (- (/ -1.0 v) (/ (* sinTheta_O sinTheta_i) v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((0.6931f + logf((0.5f / v))) + ((cosTheta_i / (v / cosTheta_O)) + ((-1.0f / v) - ((sinTheta_O * sinTheta_i) / 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(((0.6931e0 + log((0.5e0 / v))) + ((costheta_i / (v / costheta_o)) + (((-1.0e0) / v) - ((sintheta_o * sintheta_i) / v)))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(0.6931) + log((single(0.5) / v))) + ((cosTheta_i / (v / cosTheta_O)) + ((single(-1.0) / v) - ((sinTheta_O * sinTheta_i) / v)))));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Final simplification 99.6%

   \[\leadsto e^{\left(0.6931 + \log \left(\frac{0.5}{v}\right)\right) + \left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} + \left(\frac{-1}{v} - \frac{sinTheta_O \cdot sinTheta_i}{v}\right)\right)} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (pow
   E
   (+
    (- (* cosTheta_i (/ cosTheta_O v)) (* sinTheta_i (/ sinTheta_O v)))
    (+ 0.6931 (/ -1.0 v))))))

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(2.71828182845904523536) ^ (((cosTheta_i * (cosTheta_O / v)) - (sinTheta_i * (sinTheta_O / v))) + (single(0.6931) + (single(-1.0) / v))));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{v} \cdot cosTheta_O - \frac{sinTheta_i}{v} \cdot sinTheta_O\right) + \left(\frac{-1}{v} + 0.6931\right)} \cdot \frac{0.5}{v}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.6%

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

   2. exp-prod 99.6%

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

   3. *-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. +-commutative 99.6%

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

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(cosTheta_O \cdot \frac{cosTheta_i}{v} - sinTheta_O \cdot \frac{sinTheta_i}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)}} \cdot \frac{0.5}{v} \]
6. Step-by-step derivation

   1. exp-1-e 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-/r/ 99.6%

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

   4. *-rgt-identity 99.6%

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

   5. associate-*r/ 99.6%

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

   6. associate-/r/ 99.6%

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

   7. associate-*l/ 99.6%

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

   8. *-lft-identity 99.6%

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

   9. associate-*r/ 99.6%

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

   10. *-commutative 99.6%

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

   11. *-rgt-identity 99.6%

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

   12. associate-*r/ 99.6%

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

   13. associate-*l* 99.6%

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

   14. associate-*r/ 99.6%

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

   15. *-commutative 99.6%

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

   16. *-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot {e}^{\left(\left(cosTheta_i \cdot \frac{cosTheta_O}{v} - sinTheta_i \cdot \frac{sinTheta_O}{v}\right) + \left(0.6931 + \frac{-1}{v}\right)\right)} \]

Alternative 6: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1 + cosTheta_O \cdot cosTheta_i}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ -1.0 (* cosTheta_O cosTheta_i)) v)))))

C

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 + (cosTheta_O * cosTheta_i)) / 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 = (0.5e0 / v) * exp((0.6931e0 + (((-1.0e0) + (costheta_o * costheta_i)) / v)))
end function

Julia

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(Float32(-1.0) + Float32(cosTheta_O * cosTheta_i)) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) + ((single(-1.0) + (cosTheta_O * cosTheta_i)) / v)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around 0 99.6%

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

5. Taylor expanded in v around -inf 99.6%

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

   1. mul-1-neg 99.6%

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

   2. unsub-neg 99.6%

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

   3. mul-1-neg 99.6%

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

   4. unsub-neg 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1 + cosTheta_O \cdot cosTheta_i}{v}} \]

Alternative 7: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ -1.0 v)))))

C

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

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

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

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 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. exp-sum 99.6%

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

3. Simplified 99.6%

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

4. Taylor expanded in sinTheta_i around 0 99.6%

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

5. Taylor expanded in cosTheta_i around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 8: 12.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((sinTheta_O * -sinTheta_i) / 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_o * -sintheta_i) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in sinTheta_i around inf 11.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
5. Step-by-step derivation

   1. associate-*r/ 11.0%

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

   2. neg-mul-1 11.0%

      \[\leadsto e^{\frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v}} \]

   3. distribute-rgt-neg-in 11.0%

      \[\leadsto e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}} \]

6. Simplified 11.0%

   \[\leadsto e^{\color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}} \]

7. Final simplification 11.0%

   \[\leadsto e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}} \]

Alternative 9: 12.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ sinTheta_i (/ v (- sinTheta_O)))))

C

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

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 / (v / -sintheta_o)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_i / (v / -sinTheta_O)));
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in sinTheta_i around inf 11.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
5. Step-by-step derivation

   1. associate-*r/ 11.0%

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

   2. neg-mul-1 11.0%

      \[\leadsto e^{\frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v}} \]

   3. distribute-rgt-neg-in 11.0%

      \[\leadsto e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}} \]

6. Simplified 11.0%

   \[\leadsto e^{\color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 11.0%

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

   2. associate-/l* 11.0%

      \[\leadsto 1 \cdot e^{\color{blue}{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}} \]

8. Applied egg-rr 11.0%

   \[\leadsto \color{blue}{1 \cdot e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}}} \]

9. Final simplification 11.0%

   \[\leadsto e^{\frac{sinTheta_i}{\frac{v}{-sinTheta_O}}} \]

Alternative 10: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 1.0)

C

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

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 = 1.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.6%

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. Step-by-step derivation

   1. associate-+l+ 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)}} \]

   2. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   3. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} - \left(-\frac{1}{v}\right)\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   4. associate-+l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) + \left(-\frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   5. sub-neg 99.6%

      \[\leadsto e^{\color{blue}{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   6. associate--l- 99.6%

      \[\leadsto e^{\color{blue}{\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right)} + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   7. associate-/l* 99.6%

      \[\leadsto e^{\left(\color{blue}{\frac{cosTheta_i}{\frac{v}{cosTheta_O}}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{1}{2 \cdot v}\right)\right)} \]

   8. associate-/r* 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}\right)} \]

   9. metadata-eval 99.6%

      \[\leadsto e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{\color{blue}{0.5}}{v}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{e^{\left(\frac{cosTheta_i}{\frac{v}{cosTheta_O}} - \left(\frac{sinTheta_i \cdot sinTheta_O}{v} + \frac{1}{v}\right)\right) + \left(0.6931 + \log \left(\frac{0.5}{v}\right)\right)}} \]

4. Taylor expanded in sinTheta_i around inf 11.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]
5. Step-by-step derivation

   1. associate-*r/ 11.0%

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

   2. neg-mul-1 11.0%

      \[\leadsto e^{\frac{\color{blue}{-sinTheta_i \cdot sinTheta_O}}{v}} \]

   3. distribute-rgt-neg-in 11.0%

      \[\leadsto e^{\frac{\color{blue}{sinTheta_i \cdot \left(-sinTheta_O\right)}}{v}} \]

6. Simplified 11.0%

   \[\leadsto e^{\color{blue}{\frac{sinTheta_i \cdot \left(-sinTheta_O\right)}{v}}} \]

7. Taylor expanded in sinTheta_i around 0 6.4%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023238 
(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))))))