HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.8%

Time: 14.7s

Alternatives: 10

Speedup: 1.2×

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: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(2 \cdot v\right) - \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}, -0.5, 0.34655\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  (exp 2.0)
  (fma
   (- (log (* 2.0 v)) (/ (fma cosTheta_O cosTheta_i -1.0) v))
   -0.5
   0.34655)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(expf(2.0f), fmaf((logf((2.0f * v)) - (fmaf(cosTheta_O, cosTheta_i, -1.0f) / v)), -0.5f, 0.34655f));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(2.0)) ^ fma(Float32(log(Float32(Float32(2.0) * v)) - Float32(fma(cosTheta_O, cosTheta_i, Float32(-1.0)) / v)), Float32(-0.5), Float32(0.34655))
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

4. Applied egg-rr 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(v \cdot 2\right) - \frac{\mathsf{fma}\left(sinTheta\_i, -sinTheta\_O, \mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)\right)}{v}, -0.5, 0.34655\right)\right)}} \]

9. Taylor expanded in sinTheta_i around 0

   \[\leadsto {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(v \cdot 2\right) - \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}, \frac{-1}{2}, \frac{6931}{20000}\right)\right)} \]
10. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(v \cdot 2\right) - \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}, \frac{-1}{2}, \frac{6931}{20000}\right)\right)} \]

    2. sub-neg N/A

       \[\leadsto {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(v \cdot 2\right) - \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}, \frac{-1}{2}, \frac{6931}{20000}\right)\right)} \]

    3. metadata-eval N/A

       \[\leadsto {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(v \cdot 2\right) - \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}, \frac{-1}{2}, \frac{6931}{20000}\right)\right)} \]

    4. lower-fma.f32 99.7

       \[\leadsto {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(v \cdot 2\right) - \frac{\color{blue}{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}}{v}, -0.5, 0.34655\right)\right)} \]

11. Simplified 99.7%

    \[\leadsto {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(v \cdot 2\right) - \color{blue}{\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}}, -0.5, 0.34655\right)\right)} \]

12. Final simplification 99.7%

    \[\leadsto {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\log \left(2 \cdot v\right) - \frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}, -0.5, 0.34655\right)\right)} \]
13. Add Preprocessing

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in cosTheta_i around 0

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

   1. associate--l+ N/A

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

   2. associate-*r/ N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. distribute-frac-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in sinTheta_O around 0

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

   1. sub-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 99.7

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{1}{e^{\log \left(2 \cdot v\right) - \left(0.6931 + \frac{-1}{v}\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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))))))