HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 13.1s

Alternatives: 9

Speedup: 2.0×

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: 99.7% 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: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. associate--l- 99.9%

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

   3. associate-/l* 99.9%

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

   4. associate-/l* 99.9%

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

   5. associate-/r* 99.9%

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

   6. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. associate--r+ 99.9%

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

   4. associate-+l- 99.9%

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

   5. associate-*r/ 99.9%

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

   6. associate-*r/ 99.9%

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

   7. sub-div 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-1-e 99.9%

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

8. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

10. Simplified 99.9%

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

11. Taylor expanded in cosTheta_i around inf 99.9%

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

12. Taylor expanded in cosTheta_O around 0 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.9%

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

   2. *-commutative 99.9%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. fma-define 99.9%

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

   4. associate--r+ 99.9%

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

   5. associate-+l- 99.9%

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

   6. associate-*r/ 99.9%

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

   7. associate-*r/ 99.9%

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

   8. sub-div 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate-*r/ 99.9%

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

   2. associate--r+ 99.9%

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

   3. associate-/l* 99.9%

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

9. Simplified 99.9%

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

    1. *-un-lft-identity 99.9%

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

    2. exp-prod 99.9%

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

    3. e-exp-1 99.9%

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

    4. associate--l- 99.9%

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

11. Applied egg-rr 99.9%

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

12. Taylor expanded in sinTheta_O around 0 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.9%

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

   2. *-commutative 99.9%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{0.6931} - \frac{1}{v}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. e-exp-1 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * expf((0.6931f + (-1.0f / v)))) / 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 * exp((0.6931e0 + ((-1.0e0) / v)))) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.9%

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

   2. *-commutative 99.9%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. fma-define 99.9%

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

   4. associate--r+ 99.9%

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

   5. associate-+l- 99.9%

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

   6. associate-*r/ 99.9%

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

   7. associate-*r/ 99.9%

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

   8. sub-div 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in cosTheta_i around 0 99.9%

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

   1. associate-*r/ 99.9%

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

   2. associate--r+ 99.9%

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

   3. associate-/l* 99.9%

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

9. Simplified 99.9%

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

10. Taylor expanded in sinTheta_O around 0 99.9%

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

11. Final simplification 99.9%

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

Alternative 5: 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 99.9%

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

   1. exp-sum 99.9%

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

   2. *-commutative 99.9%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 6: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * expf((-1.0f / v))) / 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 * exp(((-1.0e0) / v))) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.9%

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

   2. *-commutative 99.9%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Taylor expanded in v around 0 99.2%

   \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\frac{-1}{v}}} \]
8. Step-by-step derivation

   1. associate-*l/ 99.2%

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

9. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{0.5 \cdot e^{\frac{-1}{v}}}{v}} \]
10. Add Preprocessing

Alternative 7: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.9%

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

   2. *-commutative 99.9%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Taylor expanded in v around 0 99.2%

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

Alternative 8: 13.4% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow E (* cosTheta_O (/ cosTheta_i 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)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(1)) ^ Float32(cosTheta_O * Float32(cosTheta_i / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(2.71828182845904523536) ^ (cosTheta_O * (cosTheta_i / v));
end

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. associate--l- 99.9%

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

   3. associate-/l* 99.9%

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

   4. associate-/l* 99.9%

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

   5. associate-/r* 99.9%

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

   6. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

   3. associate--r+ 99.9%

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

   4. associate-+l- 99.9%

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

   5. associate-*r/ 99.9%

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

   6. associate-*r/ 99.9%

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

   7. sub-div 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-1-e 99.9%

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

8. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

10. Simplified 99.9%

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

11. Taylor expanded in cosTheta_i around inf 99.9%

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

12. Taylor expanded in cosTheta_O around inf 13.0%

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

    1. associate-*r/ 13.0%

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

14. Simplified 13.0%

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

Alternative 9: 4.6% accurate, 74.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / 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
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / v;
end

Derivation

1. Initial program 99.9%

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

   1. exp-sum 99.9%

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

   2. *-commutative 99.9%

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

   3. rem-exp-log 99.9%

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

   4. associate-/r* 99.9%

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

   5. metadata-eval 99.9%

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

   6. associate--l- 99.9%

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

   7. associate-/l* 99.9%

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

   8. associate-/l* 99.9%

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

   9. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0 99.9%

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

6. Taylor expanded in cosTheta_O around 0 99.9%

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

7. Taylor expanded in v around 0 99.2%

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

8. Taylor expanded in v around inf 4.6%

   \[\leadsto \color{blue}{\frac{0.5}{v}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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))))))