HairBSDF, Mp, lower

Percentage Accurate: 99.5% → 99.5%

Time: 13.8s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931}}\right)}^{2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow
  (sqrt
   (*
    (pow (expm1 (log1p E)) (log (/ 0.5 v)))
    (exp
     (+
      (/ (+ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) -1.0) v)
      0.6931))))
  2.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(sqrtf((powf(expm1f(log1pf(((float) M_E))), logf((0.5f / v))) * expf((((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) + -1.0f) / v) + 0.6931f)))), 2.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return sqrt(Float32((expm1(log1p(Float32(exp(1)))) ^ log(Float32(Float32(0.5) / v))) * exp(Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) - Float32(sinTheta_i * sinTheta_O)) + Float32(-1.0)) / v) + Float32(0.6931))))) ^ Float32(2.0)
end

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-exp-log 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. exp-prod 99.7%

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

6. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{1}\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

   2. expm1-undefine 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{1}\right)} - 1\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

   3. exp-1-e 99.7%

      \[\leadsto {\left(\sqrt{{\left(e^{\mathsf{log1p}\left(\color{blue}{e}\right)} - 1\right)}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

8. Applied egg-rr 99.7%

   \[\leadsto {\left(\sqrt{{\color{blue}{\left(e^{\mathsf{log1p}\left(e\right)} - 1\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]
9. Step-by-step derivation

   1. expm1-define 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

10. Simplified 99.7%

    \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

11. Final simplification 99.7%

    \[\leadsto {\left(\sqrt{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931}}\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-exp-log 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. exp-prod 99.7%

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

6. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{1}\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

   2. expm1-undefine 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{1}\right)} - 1\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

   3. exp-1-e 99.7%

      \[\leadsto {\left(\sqrt{{\left(e^{\mathsf{log1p}\left(\color{blue}{e}\right)} - 1\right)}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

8. Applied egg-rr 99.7%

   \[\leadsto {\left(\sqrt{{\color{blue}{\left(e^{\mathsf{log1p}\left(e\right)} - 1\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]
9. Step-by-step derivation

   1. expm1-define 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

10. Simplified 99.7%

    \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

11. Taylor expanded in v around 0 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-exp-log 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. exp-prod 99.7%

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

6. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-1-e 99.7%

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

8. Applied egg-rr 99.7%

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

   1. *-lft-identity 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-exp-log 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. exp-prod 99.7%

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

6. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{1}\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

   2. expm1-undefine 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{1}\right)} - 1\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

   3. exp-1-e 99.7%

      \[\leadsto {\left(\sqrt{{\left(e^{\mathsf{log1p}\left(\color{blue}{e}\right)} - 1\right)}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

8. Applied egg-rr 99.7%

   \[\leadsto {\left(\sqrt{{\color{blue}{\left(e^{\mathsf{log1p}\left(e\right)} - 1\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]
9. Step-by-step derivation

   1. expm1-define 99.7%

      \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

10. Simplified 99.7%

    \[\leadsto {\left(\sqrt{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}}^{\log \left(\frac{0.5}{v}\right)} \cdot e^{\frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v} + 0.6931}}\right)}^{2} \]

11. Taylor expanded in sinTheta_i around 0 99.7%

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

12. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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 = sqrt(((0.5e0 / v) * exp((0.6931e0 + ((-1.0e0) / v))))) ** 2.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. exp-diff 99.3%

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

   2. +-commutative 99.3%

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

   3. exp-sum 99.3%

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

   4. rem-exp-log 99.3%

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

   5. associate-*l/ 99.3%

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

   6. +-commutative 99.3%

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

   7. *-commutative 99.3%

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

   8. fma-undefine 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in sinTheta_i around 0 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. pow2 99.3%

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

   3. associate-/l* 99.7%

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

   4. div-exp 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 6: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{2 \cdot e^{\frac{1}{v} + -0.6931}}}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (1.0f / (2.0f * expf(((1.0f / v) + -0.6931f)))) / 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 = (1.0e0 / (2.0e0 * exp(((1.0e0 / v) + (-0.6931e0))))) / v
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. exp-diff 99.3%

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

   2. +-commutative 99.3%

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

   3. exp-sum 99.3%

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

   4. rem-exp-log 99.3%

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

   5. associate-*l/ 99.3%

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

   6. +-commutative 99.3%

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

   7. *-commutative 99.3%

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

   8. fma-undefine 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in sinTheta_i around 0 99.3%

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

   1. clear-num 99.3%

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

   2. inv-pow 99.3%

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

   3. associate-*l/ 99.3%

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

8. Applied egg-rr 99.3%

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

   1. unpow-1 99.3%

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

   2. associate-/r/ 99.7%

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

10. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. associate-/r* 99.7%

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

    3. add-exp-log 99.7%

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

    4. rec-exp 99.7%

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

    5. log-div 99.7%

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

    6. add-log-exp 99.7%

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

    7. *-commutative 99.7%

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

    8. log-prod 99.7%

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

    9. rem-log-exp 99.7%

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

12. Applied egg-rr 99.7%

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

    1. *-lft-identity 99.7%

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

    2. exp-neg 99.7%

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

    3. exp-diff 99.7%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{e^{\frac{1}{v}}}{e^{0.6931 + \log 0.5}}}}}{v} \]

    4. *-rgt-identity 99.7%

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

    5. +-commutative 99.7%

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

    6. exp-sum 99.7%

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

    7. rem-exp-log 99.7%

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

    8. *-commutative 99.7%

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

    9. times-frac 99.7%

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

    10. metadata-eval 99.7%

        \[\leadsto \frac{\frac{1}{\frac{e^{\frac{1}{v}}}{e^{0.6931}} \cdot \color{blue}{2}}}{v} \]

    11. *-commutative 99.7%

        \[\leadsto \frac{\frac{1}{\color{blue}{2 \cdot \frac{e^{\frac{1}{v}}}{e^{0.6931}}}}}{v} \]

    12. div-exp 99.7%

        \[\leadsto \frac{\frac{1}{2 \cdot \color{blue}{e^{\frac{1}{v} - 0.6931}}}}{v} \]

    13. sub-neg 99.7%

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

    14. metadata-eval 99.7%

        \[\leadsto \frac{\frac{1}{2 \cdot e^{\frac{1}{v} + \color{blue}{-0.6931}}}}{v} \]

14. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{\frac{1}{2 \cdot e^{\frac{1}{v} + -0.6931}}}{v}} \]
15. Add Preprocessing

Alternative 7: 99.5% 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 99.6%

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

3. Taylor expanded in cosTheta_i around 0 99.6%

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

   1. exp-diff 99.3%

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

   2. +-commutative 99.3%

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

   3. exp-sum 99.3%

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

   4. rem-exp-log 99.3%

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

   5. associate-*l/ 99.3%

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

   6. +-commutative 99.3%

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

   7. *-commutative 99.3%

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

   8. fma-undefine 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in sinTheta_i around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. div-exp 99.7%

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

   4. sub-neg 99.7%

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

   5. distribute-neg-frac 99.7%

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

   6. metadata-eval 99.7%

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

8. Simplified 99.7%

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

Alternative 8: 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 99.6%

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

3. Taylor expanded in sinTheta_i around 0 99.6%

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

4. Taylor expanded in v around 0 97.2%

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

5. Taylor expanded in cosTheta_O around 0 97.2%

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

Alternative 9: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in sinTheta_i around 0 99.6%

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

4. Taylor expanded in v around 0 97.2%

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

5. Taylor expanded in v around inf 6.5%

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

Reproduce

herbie shell --seed 2024170 
(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))))))