HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 16.0s

Alternatives: 7

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\left({\left(\frac{e^{0.6931 + \frac{-1}{v}} \cdot 0.5}{v}\right)}^{0.25}\right)}^{2}\right)}^{2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (pow (pow (pow (/ (* (exp (+ 0.6931 (/ -1.0 v))) 0.5) v) 0.25) 2.0) 2.0))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return powf(powf(powf(((expf((0.6931f + (-1.0f / v))) * 0.5f) / v), 0.25f), 2.0f), 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 = ((((exp((0.6931e0 + ((-1.0e0) / v))) * 0.5e0) / v) ** 0.25e0) ** 2.0e0) ** 2.0e0
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return ((Float32(Float32(exp(Float32(Float32(0.6931) + Float32(Float32(-1.0) / v))) * Float32(0.5)) / v) ^ Float32(0.25)) ^ Float32(2.0)) ^ Float32(2.0)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((((exp((single(0.6931) + (single(-1.0) / v))) * single(0.5)) / v) ^ single(0.25)) ^ single(2.0)) ^ single(2.0);
end

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. associate-+r+ 99.8%

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

   2. fma-define 99.8%

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

   3. prod-exp 99.7%

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

   4. add-exp-log 99.7%

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

   5. *-commutative 99.7%

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

   6. add-sqr-sqrt 99.8%

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

   7. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

8. Applied egg-rr 99.8%

   \[\leadsto {\color{blue}{\left({\left({\left(0.5 \cdot \frac{e^{0.6931 + \frac{\left(cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O\right) - 1}{v}}}{v}\right)}^{0.25}\right)}^{2}\right)}}^{2} \]

9. Taylor expanded in sinTheta_i around 0 99.8%

   \[\leadsto {\left({\color{blue}{\left({\left(\frac{0.5 \cdot e^{\left(0.6931 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}}{v}\right)}^{0.25}\right)}}^{2}\right)}^{2} \]

10. Taylor expanded in cosTheta_O around 0 99.8%

    \[\leadsto {\left({\color{blue}{\left({\left(\frac{0.5 \cdot e^{0.6931 - \frac{1}{v}}}{v}\right)}^{0.25}\right)}}^{2}\right)}^{2} \]
11. Step-by-step derivation

    1. *-commutative 99.8%

       \[\leadsto {\left({\left({\left(\frac{\color{blue}{e^{0.6931 - \frac{1}{v}} \cdot 0.5}}{v}\right)}^{0.25}\right)}^{2}\right)}^{2} \]

12. Simplified 99.8%

    \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{0.6931 - \frac{1}{v}} \cdot 0.5}{v}\right)}^{0.25}\right)}}^{2}\right)}^{2} \]

13. Final simplification 99.8%

    \[\leadsto {\left({\left({\left(\frac{e^{0.6931 + \frac{-1}{v}} \cdot 0.5}{v}\right)}^{0.25}\right)}^{2}\right)}^{2} \]
14. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (* (exp 0.6931) (exp (/ (+ (* cosTheta_i cosTheta_O) -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) * expf((((cosTheta_i * cosTheta_O) + -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) * exp((((costheta_i * costheta_o) + (-1.0e0)) / v)))
end function

Julia

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

Derivation

1. Initial program 99.8%

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

   1. exp-sum 99.7%

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

   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. Taylor expanded in sinTheta_i around 0 99.7%

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

   1. associate--l+ 99.7%

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

   2. exp-sum 99.8%

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

   3. *-commutative 99.8%

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

   4. sub-div 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 14.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 2.0000000390829628e-24)
   (exp (/ (* sinTheta_O sinTheta_i) (- v)))
   (exp (* cosTheta_O (/ cosTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_O <= 2.0000000390829628e-24f) {
		tmp = expf(((sinTheta_O * sinTheta_i) / -v));
	} else {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (costheta_o <= 2.0000000390829628e-24) then
        tmp = exp(((sintheta_o * sintheta_i) / -v))
    else
        tmp = exp((costheta_o * (costheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_O <= Float32(2.0000000390829628e-24))
		tmp = exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)));
	else
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (cosTheta_O <= single(2.0000000390829628e-24))
		tmp = exp(((sinTheta_O * sinTheta_i) / -v));
	else
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 2.00000004e-24

   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 inf 79.1%

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

      1. associate--l+ 79.0%

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

      2. associate-*r/ 79.0%

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

      3. metadata-eval 79.0%

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

      4. +-commutative 79.0%

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

      5. times-frac 95.1%

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

      6. *-commutative 95.1%

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

   7. Simplified 95.1%

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

   8. Taylor expanded in sinTheta_i around inf 12.0%

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

   if 2.00000004e-24 < cosTheta_O

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate--l- 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in sinTheta_i around inf 77.5%

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

      1. associate--l+ 77.5%

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

      2. associate-*r/ 77.5%

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

      3. metadata-eval 77.5%

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

      4. +-commutative 77.5%

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

      5. times-frac 93.3%

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

      6. *-commutative 93.3%

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

   7. Simplified 93.3%

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

   8. Taylor expanded in cosTheta_O around inf 17.4%

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

      1. associate-*r/ 17.4%

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

   10. Simplified 17.4%

       \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 14.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 2.0000000390829628e-24)
   (exp (* sinTheta_O (/ sinTheta_i (- v))))
   (exp (* cosTheta_O (/ cosTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_O <= 2.0000000390829628e-24f) {
		tmp = expf((sinTheta_O * (sinTheta_i / -v)));
	} else {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (costheta_o <= 2.0000000390829628e-24) then
        tmp = exp((sintheta_o * (sintheta_i / -v)))
    else
        tmp = exp((costheta_o * (costheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_O <= Float32(2.0000000390829628e-24))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-v))));
	else
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (cosTheta_O <= single(2.0000000390829628e-24))
		tmp = exp((sinTheta_O * (sinTheta_i / -v)));
	else
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 2.00000004e-24

   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 inf 79.1%

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

      1. associate--l+ 79.0%

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

      2. associate-*r/ 79.0%

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

      3. metadata-eval 79.0%

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

      4. +-commutative 79.0%

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

      5. times-frac 95.1%

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

      6. *-commutative 95.1%

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

   7. Simplified 95.1%

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

   8. Taylor expanded in sinTheta_i around inf 12.0%

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

      1. mul-1-neg 12.0%

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

      2. associate-/l* 12.0%

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

      3. distribute-lft-neg-in 12.0%

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

   10. Simplified 12.0%

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

   if 2.00000004e-24 < cosTheta_O

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate--l- 99.9%

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

      3. associate-/l* 99.9%

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

      4. associate-/l* 99.9%

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

      5. associate-/r* 99.9%

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

      6. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in sinTheta_i around inf 77.5%

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

      1. associate--l+ 77.5%

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

      2. associate-*r/ 77.5%

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

      3. metadata-eval 77.5%

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

      4. +-commutative 77.5%

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

      5. times-frac 93.3%

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

      6. *-commutative 93.3%

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

   7. Simplified 93.3%

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

   8. Taylor expanded in cosTheta_O around inf 17.4%

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

      1. associate-*r/ 17.4%

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

   10. Simplified 17.4%

       \[\leadsto e^{\color{blue}{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((0.6931f + (-1.0f / v))) * (0.5f / 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((0.6931e0 + ((-1.0e0) / v))) * (0.5e0 / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. exp-sum 99.7%

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

   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. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in cosTheta_O around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 6: 13.3% accurate, 2.1× speedup

Math

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

FPCore

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

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))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around inf 78.6%

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

   1. associate--l+ 78.5%

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

   2. associate-*r/ 78.5%

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

   3. metadata-eval 78.5%

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

   4. +-commutative 78.5%

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

   5. times-frac 94.6%

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

   6. *-commutative 94.6%

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

7. Simplified 94.6%

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

8. Taylor expanded in cosTheta_O around inf 13.7%

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

9. Final simplification 13.7%

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

Alternative 7: 13.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-/l* 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-/r* 99.8%

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

   6. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around inf 78.6%

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

   1. associate--l+ 78.5%

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

   2. associate-*r/ 78.5%

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

   3. metadata-eval 78.5%

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

   4. +-commutative 78.5%

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

   5. times-frac 94.6%

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

   6. *-commutative 94.6%

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

7. Simplified 94.6%

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

8. Taylor expanded in cosTheta_O around inf 13.7%

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

   1. associate-*r/ 13.7%

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

10. Simplified 13.7%

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

Reproduce

herbie shell --seed 2024152 
(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))))))