HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 14.3s

Alternatives: 6

Speedup: N/A×

Specification

\[\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land \left(-1.5707964 \leq v \land v \leq 0.1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow (sqrt E) (* 2.0 (+ (+ 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(sqrtf(((float) M_E)), (2.0f * ((0.6931f + logf((0.5f / v))) + (-1.0f / v))));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return sqrt(Float32(exp(1))) ^ Float32(Float32(2.0) * 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 = sqrt(single(2.71828182845904523536)) ^ (single(2.0) * ((single(0.6931) + log((single(0.5) / v))) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. associate--r+ 99.7%

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

   4. associate-*r/ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. sub-div 99.7%

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

6. Applied egg-rr 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. unpow-prod-down 99.8%

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

   3. exp-1-e 99.8%

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

   4. sub-div 99.8%

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

   5. fmm-def 99.8%

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

   6. distribute-rgt-neg-out 99.8%

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

   7. exp-1-e 99.8%

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

8. Applied egg-rr 99.8%

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

   1. pow-sqr 99.7%

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

   2. +-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

   5. fmm-def 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in cosTheta_O around inf 99.7%

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

12. Taylor expanded in cosTheta_O around 0 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {e}^{\left(\frac{\left(0.6931 + \frac{-1}{v}\right) - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{2}\right)}\\ \frac{0.5}{v} \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0
         (pow
          E
          (/ (- (+ 0.6931 (/ -1.0 v)) (* sinTheta_O (/ sinTheta_i v))) 2.0))))
   (* (/ 0.5 v) (* t_0 t_0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = powf(((float) M_E), (((0.6931f + (-1.0f / v)) - (sinTheta_O * (sinTheta_i / v))) / 2.0f));
	return (0.5f / v) * (t_0 * t_0);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(exp(1)) ^ Float32(Float32(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v)) - Float32(sinTheta_O * Float32(sinTheta_i / v))) / Float32(2.0))
	return Float32(Float32(Float32(0.5) / v) * Float32(t_0 * t_0))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = single(2.71828182845904523536) ^ (((single(0.6931) + (single(-1.0) / v)) - (sinTheta_O * (sinTheta_i / v))) / single(2.0));
	tmp = (single(0.5) / v) * (t_0 * t_0);
end

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.7%

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

   2. *-commutative 99.7%

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

   3. rem-exp-log 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. fma-define 99.7%

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

   4. associate--r+ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*r/ 99.7%

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

   7. sub-div 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around 0 99.7%

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

   1. associate--r+ 99.7%

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

   2. associate-/l* 99.7%

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

9. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. exp-prod 99.7%

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

    3. e-exp-1 99.7%

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

11. Applied egg-rr 99.7%

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

    1. sqr-pow 99.7%

       \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left({e}^{\left(\frac{\left(0.6931 - \frac{1}{v}\right) - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 - \frac{1}{v}\right) - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{2}\right)}\right)} \]

13. Applied egg-rr 99.7%

    \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left({e}^{\left(\frac{\left(0.6931 - \frac{1}{v}\right) - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 - \frac{1}{v}\right) - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{2}\right)}\right)} \]

14. Final simplification 99.7%

    \[\leadsto \frac{0.5}{v} \cdot \left({e}^{\left(\frac{\left(0.6931 + \frac{-1}{v}\right) - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{2}\right)} \cdot {e}^{\left(\frac{\left(0.6931 + \frac{-1}{v}\right) - sinTheta\_O \cdot \frac{sinTheta\_i}{v}}{2}\right)}\right) \]
15. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (Float32(exp(1)) ^ 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) * (single(2.71828182845904523536) ^ (single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.7%

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

   1. exp-sum 99.7%

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

   2. *-commutative 99.7%

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

   3. rem-exp-log 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. fma-define 99.7%

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

   4. associate--r+ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*r/ 99.7%

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

   7. sub-div 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around 0 99.7%

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

   1. associate--r+ 99.7%

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

   2. associate-/l* 99.7%

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

9. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. exp-prod 99.7%

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

    3. e-exp-1 99.7%

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

11. Applied egg-rr 99.7%

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

12. Taylor expanded in sinTheta_O around 0 99.5%

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

    1. exp-to-pow 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

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 99.7%

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

   1. exp-sum 99.7%

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

   2. *-commutative 99.7%

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

   3. rem-exp-log 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. associate--l- 99.7%

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

   7. associate-/l* 99.7%

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

   8. associate-/l* 99.7%

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

   9. fma-define 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. fma-define 99.7%

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

   4. associate--r+ 99.7%

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

   5. associate-*r/ 99.7%

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

   6. associate-*r/ 99.7%

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

   7. sub-div 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in cosTheta_i around 0 99.7%

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

   1. associate--r+ 99.7%

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

   2. associate-/l* 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in sinTheta_O around 0 99.7%

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

11. Final simplification 99.7%

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

Alternative 5: 97.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in v around 0 97.6%

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

7. Taylor expanded in cosTheta_O around 0 97.6%

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

Alternative 6: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in v around 0 97.6%

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

7. Taylor expanded in v around inf 6.5%

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

Reproduce

herbie shell --seed 2024151 
(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))))))