HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 14.6s

Alternatives: 9

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: 99.7% 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: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (pow E 0.6931) (pow E (+ (log (/ 0.5 v)) (/ -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), 0.6931f) * powf(((float) M_E), (logf((0.5f / v)) + (-1.0f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32((Float32(exp(1)) ^ Float32(0.6931)) * (Float32(exp(1)) ^ Float32(log(Float32(Float32(0.5) / v)) + Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(2.71828182845904523536) ^ single(0.6931)) * (single(2.71828182845904523536) ^ (log((single(0.5) / v)) + (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. Step-by-step derivation

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. associate--r+ 99.8%

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

   4. div-inv 99.8%

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

   5. clear-num 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l/ 99.8%

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

   8. div-inv 99.8%

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

   9. clear-num 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l/ 99.8%

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

   12. sub-div 99.8%

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

5. Applied egg-rr 99.8%

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

   1. exp-1-e 99.8%

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

   2. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. associate-+l+ 99.8%

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

   2. unpow-prod-up 99.9%

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

   3. sub-div 99.9%

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

   4. cancel-sign-sub-inv 99.9%

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

   5. fma-def 99.9%

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

9. Applied egg-rr 99.9%

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

    1. sub-neg 99.9%

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

    2. fma-def 99.9%

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

    3. cancel-sign-sub-inv 99.9%

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

    4. metadata-eval 99.9%

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

11. Simplified 99.9%

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

12. Taylor expanded in sinTheta_O around 0 99.9%

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

13. Taylor expanded in cosTheta_O around 0 99.9%

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

14. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow E (+ (+ 0.6931 (log (/ 0.5 v))) (/ -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), ((0.6931f + logf((0.5f / v))) + (-1.0f / v)));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(exp(1)) ^ Float32(Float32(Float32(0.6931) + log(Float32(Float32(0.5) / v))) + Float32(Float32(-1.0) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(2.71828182845904523536) ^ ((single(0.6931) + log((single(0.5) / v))) + (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. Step-by-step derivation

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. associate--r+ 99.8%

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

   4. div-inv 99.8%

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

   5. clear-num 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*l/ 99.8%

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

   8. div-inv 99.8%

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

   9. clear-num 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*l/ 99.8%

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

   12. sub-div 99.8%

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

5. Applied egg-rr 99.8%

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

   1. exp-1-e 99.8%

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

   2. +-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in sinTheta_O around 0 99.8%

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

9. Taylor expanded in cosTheta_O around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 40.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \mathbf{if}\;sinTheta_O \leq -1.999999936531045 \cdot 10^{-19} \lor \neg \left(sinTheta_O \leq 1.4999999853326784 \cdot 10^{-10}\right):\\ \;\;\;\;e^{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* sinTheta_O (- sinTheta_i)) v)))
   (if (or (<= sinTheta_O -1.999999936531045e-19)
           (not (<= sinTheta_O 1.4999999853326784e-10)))
     (exp t_0)
     t_0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (sinTheta_O * -sinTheta_i) / v;
	float tmp;
	if ((sinTheta_O <= -1.999999936531045e-19f) || !(sinTheta_O <= 1.4999999853326784e-10f)) {
		tmp = expf(t_0);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = (sintheta_o * -sintheta_i) / v
    if ((sintheta_o <= (-1.999999936531045e-19)) .or. (.not. (sintheta_o <= 1.4999999853326784e-10))) then
        tmp = exp(t_0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(sinTheta_O * Float32(-sinTheta_i)) / v)
	tmp = Float32(0.0)
	if ((sinTheta_O <= Float32(-1.999999936531045e-19)) || !(sinTheta_O <= Float32(1.4999999853326784e-10)))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (sinTheta_O * -sinTheta_i) / v;
	tmp = single(0.0);
	if ((sinTheta_O <= single(-1.999999936531045e-19)) || ~((sinTheta_O <= single(1.4999999853326784e-10))))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < -1.99999994e-19 or 1.49999999e-10 < sinTheta_O

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in sinTheta_i around inf 22.4%

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

      1. associate-*r/ 22.4%

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

      2. neg-mul-1 22.4%

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

      3. distribute-rgt-neg-in 22.4%

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

   6. Simplified 22.4%

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

   if -1.99999994e-19 < sinTheta_O < 1.49999999e-10

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in sinTheta_i around inf 8.3%

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

      1. associate-*r/ 8.3%

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

      2. neg-mul-1 8.3%

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

      3. distribute-rgt-neg-in 8.3%

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

   6. Simplified 8.3%

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

   7. Taylor expanded in sinTheta_O around 0 6.5%

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

      1. mul-1-neg 6.5%

         \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

      2. associate-/l* 6.5%

         \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right) \]

      3. unsub-neg 6.5%

         \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

      4. associate-/r/ 6.5%

         \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

   9. Simplified 6.5%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{v} \cdot sinTheta_i} \]

   10. Taylor expanded in sinTheta_O around inf 45.9%

       \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta_O \leq -1.999999936531045 \cdot 10^{-19} \lor \neg \left(sinTheta_O \leq 1.4999999853326784 \cdot 10^{-10}\right):\\ \;\;\;\;e^{\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v}\\ \end{array} \]

Alternative 4: 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%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in sinTheta_O around 0 99.8%

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

5. Taylor expanded in cosTheta_O around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 5: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((single(-1.0) + (cosTheta_O * cosTheta_i)) / 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. Taylor expanded in v around 0 98.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i - \left(1 + sinTheta_O \cdot sinTheta_i\right)}{v}}} \]
5. Step-by-step derivation

   1. associate--r+ 98.2%

      \[\leadsto e^{\frac{\color{blue}{\left(cosTheta_O \cdot cosTheta_i - 1\right) - sinTheta_O \cdot sinTheta_i}}{v}} \]

6. Simplified 98.2%

   \[\leadsto e^{\color{blue}{\frac{\left(cosTheta_O \cdot cosTheta_i - 1\right) - sinTheta_O \cdot sinTheta_i}{v}}} \]

7. Taylor expanded in sinTheta_O around 0 98.2%

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

8. Final simplification 98.2%

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

Alternative 6: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. exp-sum 99.8%

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

   2. *-commutative 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.8%

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

   10. sub-neg 99.8%

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

   11. associate-+l+ 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in sinTheta_O around 0 99.8%

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

5. Taylor expanded in cosTheta_O around 0 99.8%

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

6. Taylor expanded in v around 0 98.3%

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

7. Final simplification 98.3%

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

Alternative 7: 20.1% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{-sinTheta_O}{\frac{v}{sinTheta_i}} \end{array} \]

FPCore

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

C

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

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 = -sintheta_o / (v / sintheta_i)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = -sinTheta_O / (v / sinTheta_i);
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. Taylor expanded in sinTheta_i around inf 14.3%

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

   1. associate-*r/ 14.3%

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

   2. neg-mul-1 14.3%

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

   3. distribute-rgt-neg-in 14.3%

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

6. Simplified 14.3%

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

7. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. mul-1-neg 6.2%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-/l* 6.2%

      \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right) \]

   3. unsub-neg 6.2%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   4. associate-/r/ 6.2%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

9. Simplified 6.2%

   \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{v} \cdot sinTheta_i} \]

10. Taylor expanded in sinTheta_O around inf 32.1%

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

    1. mul-1-neg 32.1%

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

    2. associate-/l* 14.9%

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

    3. distribute-neg-frac 14.9%

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

12. Simplified 14.9%

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

13. Final simplification 14.9%

    \[\leadsto \frac{-sinTheta_O}{\frac{v}{sinTheta_i}} \]

Alternative 8: 38.0% accurate, 37.2× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta_O \cdot \left(-sinTheta_i\right)}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (sinTheta_O * -sinTheta_i) / 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 = (sintheta_o * -sintheta_i) / v
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (sinTheta_O * -sinTheta_i) / 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. Taylor expanded in sinTheta_i around inf 14.3%

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

   1. associate-*r/ 14.3%

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

   2. neg-mul-1 14.3%

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

   3. distribute-rgt-neg-in 14.3%

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

6. Simplified 14.3%

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

7. Taylor expanded in sinTheta_O around 0 6.2%

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

   1. mul-1-neg 6.2%

      \[\leadsto 1 + \color{blue}{\left(-\frac{sinTheta_O \cdot sinTheta_i}{v}\right)} \]

   2. associate-/l* 6.2%

      \[\leadsto 1 + \left(-\color{blue}{\frac{sinTheta_O}{\frac{v}{sinTheta_i}}}\right) \]

   3. unsub-neg 6.2%

      \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{\frac{v}{sinTheta_i}}} \]

   4. associate-/r/ 6.2%

      \[\leadsto 1 - \color{blue}{\frac{sinTheta_O}{v} \cdot sinTheta_i} \]

9. Simplified 6.2%

   \[\leadsto \color{blue}{1 - \frac{sinTheta_O}{v} \cdot sinTheta_i} \]

10. Taylor expanded in sinTheta_O around inf 32.1%

    \[\leadsto \color{blue}{-1 \cdot \frac{sinTheta_O \cdot sinTheta_i}{v}} \]

11. Final simplification 32.1%

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

Alternative 9: 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. Taylor expanded in v around 0 98.2%

   \[\leadsto e^{\color{blue}{\frac{cosTheta_O \cdot cosTheta_i - \left(1 + sinTheta_O \cdot sinTheta_i\right)}{v}}} \]
5. Step-by-step derivation

   1. associate--r+ 98.2%

      \[\leadsto e^{\frac{\color{blue}{\left(cosTheta_O \cdot cosTheta_i - 1\right) - sinTheta_O \cdot sinTheta_i}}{v}} \]

6. Simplified 98.2%

   \[\leadsto e^{\color{blue}{\frac{\left(cosTheta_O \cdot cosTheta_i - 1\right) - sinTheta_O \cdot sinTheta_i}{v}}} \]

7. Taylor expanded in v around inf 6.4%

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

8. Final simplification 6.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023305 
(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))))))