HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.5%

Time: 13.7s

Alternatives: 10

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{1}{e^{cosTheta\_O \cdot \left(\frac{\left(\frac{1}{v} - \log \left(\frac{0.5}{v}\right)\right) - 0.6931}{cosTheta\_O} - \frac{cosTheta\_i}{v}\right)}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  1.0
  (exp
   (*
    cosTheta_O
    (-
     (/ (- (- (/ 1.0 v) (log (/ 0.5 v))) 0.6931) cosTheta_O)
     (/ cosTheta_i v))))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0 / exp((costheta_o * (((((1.0e0 / v) - log((0.5e0 / v))) - 0.6931e0) / costheta_o) - (costheta_i / v))))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / exp(Float32(cosTheta_O * Float32(Float32(Float32(Float32(Float32(Float32(1.0) / v) - log(Float32(Float32(0.5) / v))) - Float32(0.6931)) / cosTheta_O) - Float32(cosTheta_i / v)))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / exp((cosTheta_O * (((((single(1.0) / v) - log((single(0.5) / v))) - single(0.6931)) / cosTheta_O) - (cosTheta_i / v))));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in sinTheta_i around 0 99.5%

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

4. Taylor expanded in cosTheta_O around -inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

6. Simplified 99.5%

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

   1. distribute-rgt-neg-out 99.5%

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

   2. exp-neg 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{cosTheta\_O \cdot \left(\frac{cosTheta\_i}{v} - \frac{\frac{1}{v} + \left(-0.6931 - \log \left(\frac{0.5}{v}\right)\right)}{cosTheta\_O}\right)} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (*
   cosTheta_O
   (-
    (/ cosTheta_i v)
    (/ (+ (/ 1.0 v) (- -0.6931 (log (/ 0.5 v)))) cosTheta_O)))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 / v) - (((1.0e0 / v) + ((-0.6931e0) - log((0.5e0 / v)))) / costheta_o))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((cosTheta_O * ((cosTheta_i / v) - (((single(1.0) / v) + (single(-0.6931) - log((single(0.5) / v)))) / cosTheta_O))));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in sinTheta_i around 0 99.5%

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

4. Taylor expanded in cosTheta_O around -inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in cosTheta_O around 0 99.5%

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

   1. associate-*r/ 99.5%

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

   2. sub-neg 99.5%

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

   3. distribute-neg-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate-+r+ 99.5%

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

   6. neg-mul-1 99.5%

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

   7. distribute-neg-in 99.5%

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

   8. metadata-eval 99.5%

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

   9. unsub-neg 99.5%

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

   10. +-commutative 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in cosTheta_O around inf 99.5%

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

    1. +-commutative 99.5%

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

    2. mul-1-neg 99.5%

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

    3. unsub-neg 99.5%

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

    4. sub-neg 99.5%

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

    5. distribute-neg-in 99.5%

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

    6. metadata-eval 99.5%

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

    7. sub-neg 99.5%

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

12. Simplified 99.5%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{v} \cdot \frac{1}{{e}^{\left(\frac{1}{v} + -0.6931\right)}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (/ 1.0 (pow E (+ (/ 1.0 v) -0.6931)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * (1.0f / powf(((float) M_E), ((1.0f / v) + -0.6931f)));
}

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(1.0) / (single(2.71828182845904523536) ^ ((single(1.0) / v) + single(-0.6931))));
end

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-/l* 99.5%

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

   9. associate-/l* 99.5%

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

   10. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. associate-*r/ 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-undefine 99.5%

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

   4. associate-/l* 99.5%

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

   5. associate--l- 99.5%

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

   6. associate-+l- 99.5%

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

   7. exp-diff 83.1%

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

   8. sub-div 83.1%

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

   9. sub-neg 83.1%

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

   10. metadata-eval 83.1%

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

6. Applied egg-rr 83.1%

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

7. Taylor expanded in v around inf 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

   3. exp-1-e 99.6%

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

9. Applied egg-rr 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{v} \cdot \frac{1}{e^{\frac{1}{v} + -0.6931}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (/ 1.0 (exp (+ (/ 1.0 v) -0.6931)))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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) * (1.0e0 / exp(((1.0e0 / v) + (-0.6931e0))))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * (single(1.0) / exp(((single(1.0) / v) + single(-0.6931))));
end

Derivation

1. Initial program 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-/l* 99.5%

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

   9. associate-/l* 99.5%

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

   10. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. associate-*r/ 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-undefine 99.5%

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

   4. associate-/l* 99.5%

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

   5. associate--l- 99.5%

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

   6. associate-+l- 99.5%

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

   7. exp-diff 83.1%

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

   8. sub-div 83.1%

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

   9. sub-neg 83.1%

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

   10. metadata-eval 83.1%

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

6. Applied egg-rr 83.1%

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

7. Taylor expanded in v around inf 99.5%

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

Alternative 5: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ -1.0 v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
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 99.5%

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

   1. exp-sum 99.5%

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

   2. *-commutative 99.5%

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

   3. rem-exp-log 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate--l- 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-/l* 99.5%

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

   9. associate-/l* 99.5%

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

   10. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in sinTheta_i around 0 99.5%

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

6. Taylor expanded in cosTheta_O around 0 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{\frac{-1 + cosTheta\_O \cdot cosTheta\_i}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ (+ -1.0 (* cosTheta_O cosTheta_i)) v)))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
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 99.5%

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

3. Taylor expanded in v around 0 96.6%

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

4. Taylor expanded in sinTheta_O around 0 96.6%

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

5. Final simplification 96.6%

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

Alternative 7: 13.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{sinTheta\_i \cdot \frac{sinTheta\_O}{-v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (* sinTheta_i (/ sinTheta_O (- v)))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(sinTheta_i * Float32(sinTheta_O / Float32(-v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_i * (sinTheta_O / -v)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in sinTheta_i around inf 77.3%

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

4. Taylor expanded in sinTheta_i around inf 14.6%

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

   1. mul-1-neg 14.6%

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

   2. distribute-neg-frac2 14.6%

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

6. Simplified 14.6%

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

Alternative 8: 13.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (* sinTheta_O (/ sinTheta_i (- v)))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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_o * (sintheta_i / -v)))
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-v))))
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((sinTheta_O * (sinTheta_i / -v)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in sinTheta_i around inf 14.6%

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

   1. mul-1-neg 14.6%

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

   2. associate-*r/ 14.6%

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

   3. distribute-rgt-neg-in 14.6%

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

   4. distribute-neg-frac 14.6%

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

5. Simplified 14.6%

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

6. Final simplification 14.6%

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

Alternative 9: 13.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (* cosTheta_O (/ cosTheta_i v))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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 / v)))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp((cosTheta_O * (cosTheta_i / v)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0 96.6%

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

4. Taylor expanded in cosTheta_O around inf 11.3%

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

   1. associate-*r/ 11.3%

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

6. Simplified 11.3%

   \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
7. Add Preprocessing

Alternative 10: 6.4% accurate, 223.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ 1 \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 1.0)

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(1.0)
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0 96.6%

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

4. Taylor expanded in v around inf 6.5%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))