HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 13.5s

Alternatives: 7

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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-log.f32 N/A

      \[\leadsto e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   6. lift-/.f32 N/A

      \[\leadsto e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   7. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{2 \cdot v}} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{2 \cdot v}} \]

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}}}{v \cdot 2}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{v \cdot 2}{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}}}}} \]

   3. rem-exp-log N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{\log \left(v \cdot 2\right)}}}{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}}}} \]

   4. lift-log.f32 N/A

      \[\leadsto \frac{1}{\frac{e^{\color{blue}{\log \left(v \cdot 2\right)}}}{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}}}} \]

   5. lift-exp.f32 N/A

      \[\leadsto \frac{1}{\frac{e^{\log \left(v \cdot 2\right)}}{\color{blue}{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}}}}} \]

   6. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{e^{\log \left(v \cdot 2\right) - \left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}\right)}}} \]

   7. lift--.f32 N/A

      \[\leadsto \frac{1}{e^{\color{blue}{\log \left(v \cdot 2\right) - \left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}\right)}}} \]

   8. rec-exp N/A

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(\left(\log \left(v \cdot 2\right) - \left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}\right)\right)\right)}} \]

   9. neg-mul-1 N/A

      \[\leadsto e^{\color{blue}{-1 \cdot \left(\log \left(v \cdot 2\right) - \left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}\right)\right)}} \]

   10. exp-prod N/A

       \[\leadsto \color{blue}{{\left(e^{-1}\right)}^{\left(\log \left(v \cdot 2\right) - \left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}\right)\right)}} \]

   11. lower-pow.f32 N/A

       \[\leadsto \color{blue}{{\left(e^{-1}\right)}^{\left(\log \left(v \cdot 2\right) - \left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1}{v} + \frac{6931}{10000}\right)\right)}} \]

   12. lower-exp.f32 99.7

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

6. Applied rewrites 99.7%

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

   1. lift--.f32 N/A

      \[\leadsto {\left(e^{-1}\right)}^{\color{blue}{\left(\left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right) - \frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, \mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), -1\right)\right)}{v}\right)}} \]

   2. sub-neg N/A

      \[\leadsto {\left(e^{-1}\right)}^{\color{blue}{\left(\left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right) + \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, \mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), -1\right)\right)}{v}\right)\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto {\left(e^{-1}\right)}^{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, \mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), -1\right)\right)}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)}} \]

   4. lift-/.f32 N/A

      \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, \mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), -1\right)\right)}{v}}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   5. div-inv N/A

      \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, \mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), -1\right)\right) \cdot \frac{1}{v}}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   6. lift-fma.f32 N/A

      \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\color{blue}{\left(cosTheta\_i \cdot cosTheta\_O + \mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), -1\right)\right)} \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   7. lift-fma.f32 N/A

      \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\left(cosTheta\_i \cdot cosTheta\_O + \color{blue}{\left(sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right) + -1\right)}\right) \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   8. associate-+r+ N/A

      \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_O + sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right)\right) + -1\right)} \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\left(\color{blue}{\left(sinTheta\_i \cdot \left(\mathsf{neg}\left(sinTheta\_O\right)\right) + cosTheta\_i \cdot cosTheta\_O\right)} + -1\right) \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   10. lift-fma.f32 N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\left(\color{blue}{\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right)} + -1\right) \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   11. lift-*.f32 N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), \color{blue}{cosTheta\_i \cdot cosTheta\_O}\right) + -1\right) \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   13. sub-neg N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right) - 1\right)} \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   14. lift--.f32 N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right) - 1\right)} \cdot \frac{1}{v}\right)\right) + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

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

       \[\leadsto {\left(e^{-1}\right)}^{\left(\color{blue}{\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right) - 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   16. neg-mul-1 N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right) - 1\right) \cdot \color{blue}{\left(-1 \cdot \frac{1}{v}\right)} + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   17. div-inv N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right) - 1\right) \cdot \color{blue}{\frac{-1}{v}} + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

   18. lift-/.f32 N/A

       \[\leadsto {\left(e^{-1}\right)}^{\left(\left(\mathsf{fma}\left(sinTheta\_i, \mathsf{neg}\left(sinTheta\_O\right), cosTheta\_i \cdot cosTheta\_O\right) - 1\right) \cdot \color{blue}{\frac{-1}{v}} + \left(\log \left(v \cdot 2\right) - \frac{6931}{10000}\right)\right)} \]

8. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((fmaf(sinTheta_O, -sinTheta_i, -1.0f) / v) + 0.6931f)) / (v * 2.0f);
}

Julia

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

Derivation

1. Initial program 99.3%

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

   1. lift-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right) + \log \left(\frac{1}{2 \cdot v}\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot e^{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   4. lift-log.f32 N/A

      \[\leadsto e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot e^{\color{blue}{\log \left(\frac{1}{2 \cdot v}\right)}} \]

   5. rem-exp-log N/A

      \[\leadsto e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   6. lift-/.f32 N/A

      \[\leadsto e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}} \cdot \color{blue}{\frac{1}{2 \cdot v}} \]

   7. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{2 \cdot v}} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}}{2 \cdot v}} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around 0

   \[\leadsto \frac{e^{\frac{\color{blue}{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) - 1}}{v} + \frac{6931}{10000}}}{v \cdot 2} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right) + \left(\mathsf{neg}\left(1\right)\right)}}{v} + \frac{6931}{10000}}}{v \cdot 2} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\left(\mathsf{neg}\left(sinTheta\_O \cdot sinTheta\_i\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)}{v} + \frac{6931}{10000}}}{v \cdot 2} \]

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

      \[\leadsto \frac{e^{\frac{\color{blue}{sinTheta\_O \cdot \left(\mathsf{neg}\left(sinTheta\_i\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)}{v} + \frac{6931}{10000}}}{v \cdot 2} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \color{blue}{\left(-1 \cdot sinTheta\_i\right)} + \left(\mathsf{neg}\left(1\right)\right)}{v} + \frac{6931}{10000}}}{v \cdot 2} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{\frac{sinTheta\_O \cdot \left(-1 \cdot sinTheta\_i\right) + \color{blue}{-1}}{v} + \frac{6931}{10000}}}{v \cdot 2} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, -1 \cdot sinTheta\_i, -1\right)}}{v} + \frac{6931}{10000}}}{v \cdot 2} \]

   7. mul-1-neg N/A

      \[\leadsto \frac{e^{\frac{\mathsf{fma}\left(sinTheta\_O, \color{blue}{\mathsf{neg}\left(sinTheta\_i\right)}, -1\right)}{v} + \frac{6931}{10000}}}{v \cdot 2} \]

   8. lower-neg.f32 99.7

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

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) + 0.6931f));
}

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) + 0.6931e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]
4. Step-by-step derivation

   1. lower-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]

   2. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} - \frac{1}{v}} \]

   3. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}} \]

   4. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)} \]

   5. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   6. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   7. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   8. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   9. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   10. lower-/.f32 N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   11. lower-+.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   12. div-sub N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   13. lower-/.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   14. sub-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}\right)} \]

   16. lower-fma.f32 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in cosTheta_i around 0

   \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{1}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 4: 99.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}{v} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]
4. Step-by-step derivation

   1. lower-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]

   2. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} - \frac{1}{v}} \]

   3. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}} \]

   4. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)} \]

   5. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   6. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   7. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   8. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   9. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   10. lower-/.f32 N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   11. lower-+.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   12. div-sub N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   13. lower-/.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   14. sub-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}\right)} \]

   16. lower-fma.f32 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in v around 0

   \[\leadsto \frac{e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}}{v} \]
9. Step-by-step derivation

10. Applied rewrites 99.0%

    \[\leadsto \frac{e^{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}}{v} \]
11. Add Preprocessing

Alternative 5: 97.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ e^{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]
4. Step-by-step derivation

   1. lower-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]

   2. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} - \frac{1}{v}} \]

   3. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}} \]

   4. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)} \]

   5. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   6. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   7. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   8. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   9. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   10. lower-/.f32 N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   11. lower-+.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   12. div-sub N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   13. lower-/.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   14. sub-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}\right)} \]

   16. lower-fma.f32 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in v around 0

   \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}} \]
7. Step-by-step derivation

8. Applied rewrites 97.2%

   \[\leadsto e^{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}} \]
9. Add Preprocessing

Alternative 6: 13.5% accurate, 2.3× speedup

Math

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

FPCore

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

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))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]
4. Step-by-step derivation

   1. lower-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]

   2. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} - \frac{1}{v}} \]

   3. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}} \]

   4. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)} \]

   5. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   6. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   7. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   8. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   9. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   10. lower-/.f32 N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   11. lower-+.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   12. div-sub N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   13. lower-/.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   14. sub-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}\right)} \]

   16. lower-fma.f32 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in cosTheta_O around inf

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

8. Applied rewrites 15.1%

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

9. Final simplification 15.1%

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

Alternative 7: 13.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((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 = exp((costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]
4. Step-by-step derivation

   1. lower-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]

   2. associate-+r+ N/A

      \[\leadsto e^{\color{blue}{\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)} - \frac{1}{v}} \]

   3. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)}} \]

   4. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)} \]

   5. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   6. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   7. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   8. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   9. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   10. lower-/.f32 N/A

       \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   11. lower-+.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   12. div-sub N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   13. lower-/.f32 N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   14. sub-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}\right)} \]

   16. lower-fma.f32 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in cosTheta_O around inf

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

8. Applied rewrites 15.1%

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

10. Applied rewrites 15.1%

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

11. Final simplification 15.1%

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

Reproduce

herbie shell --seed 2024219 
(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))))))