HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 13.2s

Alternatives: 5

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(exp(Float32(0.6931)) / Float32(v * (exp(fma(sinTheta_O, sinTheta_i, Float32(1.0))) ^ Float32(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. Step-by-step derivation

   1. associate-+l+ 99.3%

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

   2. associate--l- 99.3%

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

   3. associate-/l* 99.3%

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

   4. associate-/l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in cosTheta_i around 0 99.3%

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

   1. exp-diff 99.1%

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

   2. +-commutative 99.1%

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

   3. associate-/l* 99.1%

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

   4. fma-undefine 99.1%

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

   5. frac-2neg 99.1%

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

   6. prod-exp 99.1%

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

   7. add-exp-log 99.1%

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

7. Applied egg-rr 99.1%

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

   1. associate-*r/ 99.1%

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

   2. associate-*l/ 98.7%

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

   3. distribute-rgt-neg-in 98.7%

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

   4. metadata-eval 98.7%

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

   5. fma-define 98.7%

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

   6. +-commutative 98.7%

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

   7. associate-*r/ 98.7%

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

   8. associate-*r/ 98.7%

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

   9. *-lft-identity 98.7%

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

   10. metadata-eval 98.7%

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

   11. cancel-sign-sub-inv 98.7%

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

   12. associate-*r/ 98.7%

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

   13. associate-*r/ 98.7%

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

   14. div-sub 98.8%

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

   15. neg-mul-1 98.8%

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

   16. distribute-lft-neg-in 98.8%

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

   17. cancel-sign-sub 98.8%

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

   18. +-commutative 98.8%

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

   19. fma-define 98.8%

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

9. Simplified 98.8%

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

10. Taylor expanded in v around 0 99.6%

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

    1. div-inv 99.5%

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

    2. exp-prod 99.6%

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

    3. +-commutative 99.6%

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

    4. fma-undefine 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. exp-sum 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. associate--l- 99.4%

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

   12. associate-+r- 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

6. Taylor expanded in cosTheta_O around 0 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((single(2.71828182845904523536) ^ (single(-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. Step-by-step derivation

   1. +-commutative 99.3%

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

   2. exp-sum 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. associate--l- 99.4%

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

   12. associate-+r- 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

6. Taylor expanded in cosTheta_O around 0 99.6%

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

7. Taylor expanded in v around 0 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. exp-prod 98.6%

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

   3. e-exp-1 98.6%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 4: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. exp-sum 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. associate--l- 99.4%

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

   12. associate-+r- 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

6. Taylor expanded in cosTheta_O around 0 99.6%

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

7. Taylor expanded in v around 0 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 4.6% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * (single(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. Step-by-step derivation

   1. +-commutative 99.3%

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

   2. exp-sum 99.4%

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

   3. rem-exp-log 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-rgt-identity 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.4%

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

   10. +-commutative 99.4%

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

   11. associate--l- 99.4%

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

   12. associate-+r- 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in sinTheta_i around 0 99.6%

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

6. Taylor expanded in cosTheta_O around 0 99.6%

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

7. Taylor expanded in v around 0 98.6%

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

8. Taylor expanded in v around inf 4.6%

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

9. Final simplification 4.6%

   \[\leadsto 0.5 \cdot \frac{1}{v} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))))