HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 14.3s

Alternatives: 11

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((((((costheta_i * costheta_o) / v) - ((sintheta_i * sintheta_o) / v)) - (1.0e0 / v)) + 0.6931e0) + log((1.0e0 / (2.0e0 * v)))))
end function

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   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. associate-+l+ N/A

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

   4. lift--.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. sub-div N/A

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

   9. lift-/.f32 N/A

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

   10. frac-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. lift-/.f32 N/A

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

   15. lift-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift--.f32 N/A

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

   4. associate-+r- N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. log-prod N/A

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

   8. associate--r+ N/A

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

   9. exp-diff N/A

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

9. Applied rewrites 99.7%

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

    1. lift--.f32 N/A

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

    2. lift-+.f32 N/A

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

    3. +-commutative N/A

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

    4. associate--l+ N/A

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

    5. lift-/.f32 N/A

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

    6. div-inv N/A

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

    7. lift-/.f32 N/A

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

    8. lower-fma.f32 N/A

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

    9. lower--.f32 99.8

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot e^{\mathsf{fma}\left(cosTheta\_O, \frac{cosTheta\_i}{v}, 0.6931\right) + \left(\frac{-1}{v} - \log v\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   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. associate-+l+ N/A

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

   4. lift--.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. sub-div N/A

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

   9. lift-/.f32 N/A

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

   10. frac-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. lift-/.f32 N/A

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

   15. lift-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift--.f32 N/A

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

   4. associate-+r- N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. log-prod N/A

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

   8. associate--r+ N/A

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

   9. exp-diff N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in sinTheta_i around 0

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

    1. lower-*.f32 N/A

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

    2. lower-exp.f32 N/A

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

    3. lower--.f32 N/A

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

    4. +-commutative N/A

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

    5. associate-/l* N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. lower-+.f32 N/A

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

    9. lower-log.f32 N/A

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

    10. lower-/.f32 99.7

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

12. Applied rewrites 99.7%

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

13. Final simplification 99.7%

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

Reproduce

herbie shell --seed 2024227 
(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))))))