HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 15.4s

Alternatives: 6

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: 99.6% 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, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (pow E (- 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 * (powf(((float) M_E), (0.6931f - (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(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) * ((single(2.71828182845904523536) ^ (single(0.6931) - (single(1.0) / v))) / v);
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. exp-sum 99.7%

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

   3. +-commutative 99.7%

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

   4. exp-sum 99.7%

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

   5. add-exp-log 99.7%

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

   6. associate-/r/ 99.7%

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

   7. associate-/r/ 99.7%

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

   8. fma-def 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around 0 99.7%

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

   1. prod-exp 99.7%

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

   2. sub-neg 99.7%

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

   3. associate-/l* 99.7%

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

9. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. exp-prod 99.7%

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

    3. exp-1-e 99.7%

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

    4. +-commutative 99.7%

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

    5. div-inv 99.7%

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

    6. clear-num 99.7%

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

    7. fma-def 99.7%

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

11. Applied egg-rr 99.7%

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

12. Taylor expanded in sinTheta_O around 0 99.5%

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

    1. exp-to-pow 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

    \[\leadsto 0.5 \cdot \frac{{e}^{\left(0.6931 - \frac{1}{v}\right)}}{v} \]
16. Add Preprocessing

Alternative 2: 13.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{-sinTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -1.000000046701102e-34)
   (exp (* sinTheta_i (/ sinTheta_O v)))
   (exp (* sinTheta_O (/ (- sinTheta_i) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= -1.000000046701102e-34f) {
		tmp = expf((sinTheta_i * (sinTheta_O / v)));
	} else {
		tmp = expf((sinTheta_O * (-sinTheta_i / v)));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (sintheta_i <= (-1.000000046701102e-34)) then
        tmp = exp((sintheta_i * (sintheta_o / v)))
    else
        tmp = exp((sintheta_o * (-sintheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-1.000000046701102e-34))
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / v)));
	else
		tmp = exp(Float32(sinTheta_O * Float32(Float32(-sinTheta_i) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-1.000000046701102e-34))
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	else
		tmp = exp((sinTheta_O * (-sinTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -1.00000005e-34

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in v around 0 95.6%

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

      1. +-commutative 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in sinTheta_O around inf 12.3%

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

      1. mul-1-neg 12.3%

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

      2. associate-/l* 12.3%

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

      3. distribute-neg-frac 12.3%

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

   10. Simplified 12.3%

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

       1. associate-/r/ 12.3%

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

       2. add-sqr-sqrt 9.6%

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

       3. sqrt-unprod 20.1%

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

       4. sqr-neg 20.1%

          \[\leadsto e^{\frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v} \cdot sinTheta\_i} \]

       5. sqrt-unprod 10.5%

          \[\leadsto e^{\frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v} \cdot sinTheta\_i} \]

       6. add-sqr-sqrt 13.9%

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

   12. Applied egg-rr 13.9%

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

   if -1.00000005e-34 < sinTheta_i

   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 v around 0 98.2%

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

      1. +-commutative 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in sinTheta_O around inf 13.5%

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

      1. mul-1-neg 13.5%

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

      2. associate-/l* 13.5%

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

      3. distribute-neg-frac 13.5%

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

   10. Simplified 13.5%

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

   11. Taylor expanded in sinTheta_O around inf 13.5%

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

       1. mul-1-neg 13.5%

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

       2. associate-*r/ 13.5%

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

       3. distribute-lft-neg-in 13.5%

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

   13. Simplified 13.5%

       \[\leadsto \color{blue}{e^{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{-sinTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. exp-sum 99.7%

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

   3. +-commutative 99.7%

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

   4. exp-sum 99.7%

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

   5. add-exp-log 99.7%

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

   6. associate-/r/ 99.7%

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

   7. associate-/r/ 99.7%

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

   8. fma-def 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around 0 99.7%

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

   1. prod-exp 99.7%

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

   2. sub-neg 99.7%

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

   3. associate-/l* 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in sinTheta_O around 0 99.7%

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

11. Final simplification 99.7%

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

Alternative 4: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{cosTheta\_O \cdot cosTheta\_i + -1}{v}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((cosTheta_O * cosTheta_i) + -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((((costheta_o * costheta_i) + (-1.0e0)) / v))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in v around 0 97.0%

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

   1. +-commutative 97.0%

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

7. Simplified 97.0%

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

8. Taylor expanded in sinTheta_O around 0 97.0%

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

9. Final simplification 97.0%

   \[\leadsto e^{\frac{cosTheta\_O \cdot cosTheta\_i + -1}{v}} \]
10. Add Preprocessing

Alternative 5: 13.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in v around 0 97.0%

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

   1. +-commutative 97.0%

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

7. Simplified 97.0%

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

8. Taylor expanded in sinTheta_O around inf 12.9%

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

   1. mul-1-neg 12.9%

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

   2. associate-/l* 12.9%

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

   3. distribute-neg-frac 12.9%

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

10. Simplified 12.9%

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

    1. associate-/r/ 12.9%

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

    2. add-sqr-sqrt 6.1%

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

    3. sqrt-unprod 12.9%

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

    4. sqr-neg 12.9%

       \[\leadsto e^{\frac{\sqrt{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}}{v} \cdot sinTheta\_i} \]

    5. sqrt-unprod 6.8%

       \[\leadsto e^{\frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{v} \cdot sinTheta\_i} \]

    6. add-sqr-sqrt 12.5%

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

12. Applied egg-rr 12.5%

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

13. Final simplification 12.5%

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

Alternative 6: 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 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in v around 0 97.0%

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

   1. +-commutative 97.0%

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

7. Simplified 97.0%

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

8. Taylor expanded in sinTheta_O around inf 12.9%

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

   1. mul-1-neg 12.9%

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

   2. associate-/l* 12.9%

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

   3. distribute-neg-frac 12.9%

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

10. Simplified 12.9%

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

11. Taylor expanded in sinTheta_O around 0 6.5%

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

12. Final simplification 6.5%

    \[\leadsto 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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))))))