HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 15.7s

Alternatives: 8

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3}\right)}^{2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  (pow
   (cbrt
    (sqrt
     (exp
      (+
       (fma cosTheta_O (/ cosTheta_i v) (log (/ 0.5 v)))
       (+ 0.6931 (/ -1.0 v))))))
   3.0)
  2.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(powf(cbrtf(sqrtf(expf((fmaf(cosTheta_O, (cosTheta_i / v), logf((0.5f / v))) + (0.6931f + (-1.0f / v)))))), 3.0f), 2.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return (cbrt(sqrt(exp(Float32(fma(cosTheta_O, Float32(cosTheta_i / v), log(Float32(Float32(0.5) / v))) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))))) ^ Float32(3.0)) ^ Float32(2.0)
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. associate--l+ 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l/ 99.8%

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

   7. *-commutative 99.8%

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

   8. fma-def 99.8%

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

7. Applied egg-rr 99.8%

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

   1. add-cube-cbrt 99.8%

      \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\sqrt{e^{0.6931 + \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}}} \cdot \sqrt[3]{\sqrt{e^{0.6931 + \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}}}\right) \cdot \sqrt[3]{\sqrt{e^{0.6931 + \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}}}\right)}}^{2} \]

   2. pow3 99.8%

      \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{e^{0.6931 + \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right)}}}\right)}^{3}\right)}}^{2} \]

   3. +-commutative 99.8%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt{e^{\color{blue}{\left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) - \frac{1}{v}\right) + 0.6931}}}}\right)}^{3}\right)}^{2} \]

   4. sub-neg 99.8%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt{e^{\color{blue}{\left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(-\frac{1}{v}\right)\right)} + 0.6931}}}\right)}^{3}\right)}^{2} \]

   5. associate-+l+ 99.8%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt{e^{\color{blue}{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(\left(-\frac{1}{v}\right) + 0.6931\right)}}}}\right)}^{3}\right)}^{2} \]

   6. distribute-neg-frac 99.8%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(\color{blue}{\frac{-1}{v}} + 0.6931\right)}}}\right)}^{3}\right)}^{2} \]

   7. metadata-eval 99.8%

      \[\leadsto {\left({\left(\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(\frac{\color{blue}{-1}}{v} + 0.6931\right)}}}\right)}^{3}\right)}^{2} \]

9. Applied egg-rr 99.8%

   \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(\frac{-1}{v} + 0.6931\right)}}}\right)}^{3}\right)}}^{2} \]

10. Final simplification 99.8%

    \[\leadsto {\left({\left(\sqrt[3]{\sqrt{e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(0.6931 + \frac{-1}{v}\right)}}}\right)}^{3}\right)}^{2} \]
11. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left({\left({\left(e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{0.25}\right)}^{2}\right)}^{2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  (pow
   (pow
    (exp
     (+
      (fma cosTheta_O (/ cosTheta_i v) (log (/ 0.5 v)))
      (+ 0.6931 (/ -1.0 v))))
    0.25)
   2.0)
  2.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(powf(powf(expf((fmaf(cosTheta_O, (cosTheta_i / v), logf((0.5f / v))) + (0.6931f + (-1.0f / v)))), 0.25f), 2.0f), 2.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return ((exp(Float32(fma(cosTheta_O, Float32(cosTheta_i / v), log(Float32(Float32(0.5) / v))) + Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)))) ^ Float32(0.25)) ^ Float32(2.0)) ^ Float32(2.0)
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. associate--l+ 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l/ 99.8%

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

   7. *-commutative 99.8%

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

   8. fma-def 99.8%

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

7. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. pow1/2 99.8%

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

   4. sqrt-pow1 99.8%

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

   5. +-commutative 99.8%

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

   6. sub-neg 99.8%

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

   7. associate-+l+ 99.8%

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

   8. distribute-neg-frac 99.8%

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

   9. metadata-eval 99.8%

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

   10. metadata-eval 99.8%

       \[\leadsto {\left({\left({\left(e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(\frac{-1}{v} + 0.6931\right)}\right)}^{\color{blue}{0.25}}\right)}^{2}\right)}^{2} \]

9. Applied egg-rr 99.8%

   \[\leadsto {\color{blue}{\left({\left({\left(e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(\frac{-1}{v} + 0.6931\right)}\right)}^{0.25}\right)}^{2}\right)}}^{2} \]

10. Final simplification 99.8%

    \[\leadsto {\left({\left({\left(e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, \log \left(\frac{0.5}{v}\right)\right) + \left(0.6931 + \frac{-1}{v}\right)}\right)}^{0.25}\right)}^{2}\right)}^{2} \]
11. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{e^{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) + \frac{-1}{v}}}\right)}^{2} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = sqrt(exp(((log((0.5e0 / v)) + 0.6931e0) + ((-1.0e0) / v)))) ** 2.0e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = sqrt(exp(((log((single(0.5) / v)) + single(0.6931)) + (single(-1.0) / v)))) ^ single(2.0);
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. associate--l+ 99.8%

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

   4. +-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l/ 99.8%

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

   7. *-commutative 99.8%

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

   8. fma-def 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in cosTheta_O around 0 99.8%

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

9. Final simplification 99.8%

   \[\leadsto {\left(\sqrt{e^{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) + \frac{-1}{v}}}\right)}^{2} \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Final simplification 99.8%

   \[\leadsto e^{\left(\log \left(\frac{0.5}{v}\right) + 0.6931\right) + \frac{-1}{v}} \]
8. 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 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

   1. exp-diff 99.6%

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

   2. +-commutative 99.6%

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

   3. exp-sum 99.6%

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

   4. rem-exp-log 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in v around 0 99.6%

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

    1. associate-*r/ 99.6%

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

    2. times-frac 99.6%

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

    3. exp-diff 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 6: 97.9% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow E (/ -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), (-1.0f / v));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in v around 0 98.2%

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

7. Taylor expanded in cosTheta_O around 0 98.2%

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

   1. *-un-lft-identity 98.2%

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

   2. exp-prod 98.2%

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

9. Applied egg-rr 98.2%

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

    1. exp-1-e 98.2%

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

11. Simplified 98.2%

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

12. Final simplification 98.2%

    \[\leadsto {e}^{\left(\frac{-1}{v}\right)} \]
13. Add Preprocessing

Alternative 7: 97.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 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 = exp(((-1.0e0) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in v around 0 98.2%

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

7. Taylor expanded in cosTheta_O around 0 98.2%

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

8. Final simplification 98.2%

   \[\leadsto e^{\frac{-1}{v}} \]
9. Add Preprocessing

Alternative 8: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around inf 18.4%

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

   1. mul-1-neg 18.4%

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

   2. associate-*r/ 18.4%

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

   3. distribute-rgt-neg-in 18.4%

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

7. Simplified 18.4%

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

8. Taylor expanded in sinTheta_O around 0 6.4%

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

9. Final simplification 6.4%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))