HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.6%

Time: 19.7s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931}}}\right)}^{3}\right)}^{3} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (pow
   (pow
    (cbrt
     (cbrt
      (exp
       (+
        (/ (+ (fma cosTheta_i cosTheta_O (* sinTheta_i sinTheta_O)) -1.0) v)
        0.6931))))
    3.0)
   3.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(powf(cbrtf(cbrtf(expf((((fmaf(cosTheta_i, cosTheta_O, (sinTheta_i * sinTheta_O)) + -1.0f) / v) + 0.6931f)))), 3.0f), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * ((cbrt(cbrt(exp(Float32(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(sinTheta_i * sinTheta_O)) + Float32(-1.0)) / v) + Float32(0.6931))))) ^ Float32(3.0)) ^ Float32(3.0)))
end

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-sum 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.8%

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

   10. +-commutative 99.8%

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

   11. associate--l- 99.8%

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

   12. associate-+r- 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.8%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}}\right)} \]

   2. pow3 99.8%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}}\right)}^{3}} \]

   3. fma-define 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)}}}\right)}^{3} \]

   4. associate--l+ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{0.6931 + \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right)}}}\right)}^{3} \]

   5. +-commutative 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931}}}\right)}^{3} \]

   6. associate--r+ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + 0.6931}}\right)}^{3} \]

   7. associate-*r/ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}}\right)}^{3} \]

   8. associate-*r/ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) - \frac{1}{v}\right) + 0.6931}}\right)}^{3} \]

   9. sub-div 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + 0.6931}}\right)}^{3} \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}\right)}^{3}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}} \cdot \sqrt[3]{\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}}\right)}}^{3} \]

   2. pow3 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}}\right)}^{3}\right)}}^{3} \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{0.5}{v} \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931}}}\right)}^{3}\right)}}^{3} \]

9. Final simplification 99.8%

   \[\leadsto \frac{0.5}{v} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v} + 0.6931}}}\right)}^{3}\right)}^{3} \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\frac{-1 + sinTheta\_i \cdot sinTheta\_O}{v}} \cdot e^{0.6931}}\right)}^{3} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (/ 0.5 v)
  (pow
   (cbrt (* (exp (/ (+ -1.0 (* sinTheta_i sinTheta_O)) v)) (exp 0.6931)))
   3.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f / v) * powf(cbrtf((expf(((-1.0f + (sinTheta_i * sinTheta_O)) / v)) * expf(0.6931f))), 3.0f);
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) / v) * (cbrt(Float32(exp(Float32(Float32(Float32(-1.0) + Float32(sinTheta_i * sinTheta_O)) / v)) * exp(Float32(0.6931)))) ^ Float32(3.0)))
end

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-sum 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.8%

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

   10. +-commutative 99.8%

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

   11. associate--l- 99.8%

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

   12. associate-+r- 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.8%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}} \cdot \sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}}\right) \cdot \sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}}\right)} \]

   2. pow3 99.8%

      \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \mathsf{fma}\left(sinTheta\_i, \frac{sinTheta\_O}{v}, \frac{1}{v}\right)}}\right)}^{3}} \]

   3. fma-define 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(0.6931 + cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right) - \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)}}}\right)}^{3} \]

   4. associate--l+ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{0.6931 + \left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right)}}}\right)}^{3} \]

   5. +-commutative 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v} + \frac{1}{v}\right)\right) + 0.6931}}}\right)}^{3} \]

   6. associate--r+ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\color{blue}{\left(\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right)} + 0.6931}}\right)}^{3} \]

   7. associate-*r/ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}} - sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931}}\right)}^{3} \]

   8. associate-*r/ 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \color{blue}{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\right) - \frac{1}{v}\right) + 0.6931}}\right)}^{3} \]

   9. sub-div 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\left(\color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v}} - \frac{1}{v}\right) + 0.6931}}\right)}^{3} \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{0.5}{v} \cdot \color{blue}{{\left(\sqrt[3]{e^{\left(\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}\right) + 0.6931}}\right)}^{3}} \]
7. Step-by-step derivation

   1. exp-sum 99.8%

      \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\color{blue}{e^{\frac{cosTheta\_i \cdot cosTheta\_O - sinTheta\_i \cdot sinTheta\_O}{v} - \frac{1}{v}} \cdot e^{0.6931}}}\right)}^{3} \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\color{blue}{e^{\frac{\mathsf{fma}\left(cosTheta\_i, cosTheta\_O, sinTheta\_i \cdot sinTheta\_O\right) + -1}{v}} \cdot e^{0.6931}}}\right)}^{3} \]

9. Taylor expanded in cosTheta_i around 0 99.8%

   \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{\color{blue}{e^{\frac{sinTheta\_O \cdot sinTheta\_i - 1}{v}}} \cdot e^{0.6931}}\right)}^{3} \]

10. Final simplification 99.8%

    \[\leadsto \frac{0.5}{v} \cdot {\left(\sqrt[3]{e^{\frac{-1 + sinTheta\_i \cdot sinTheta\_O}{v}} \cdot e^{0.6931}}\right)}^{3} \]
11. Add Preprocessing

Alternative 3: 40.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{elif}\;sinTheta\_i \leq 8.00000025099516 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* sinTheta_i sinTheta_O) v)))
   (if (<= sinTheta_i -2.499999990010493e-13)
     (exp t_0)
     (if (<= sinTheta_i 8.00000025099516e-22)
       t_0
       (exp (* sinTheta_O (/ sinTheta_i (- v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (sinTheta_i * sinTheta_O) / v;
	float tmp;
	if (sinTheta_i <= -2.499999990010493e-13f) {
		tmp = expf(t_0);
	} else if (sinTheta_i <= 8.00000025099516e-22f) {
		tmp = t_0;
	} else {
		tmp = expf((sinTheta_O * (sinTheta_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) :: t_0
    real(4) :: tmp
    t_0 = (sintheta_i * sintheta_o) / v
    if (sintheta_i <= (-2.499999990010493e-13)) then
        tmp = exp(t_0)
    else if (sintheta_i <= 8.00000025099516e-22) then
        tmp = t_0
    else
        tmp = exp((sintheta_o * (sintheta_i / -v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(sinTheta_i * sinTheta_O) / v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-2.499999990010493e-13))
		tmp = exp(t_0);
	elseif (sinTheta_i <= Float32(8.00000025099516e-22))
		tmp = t_0;
	else
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (sinTheta_i * sinTheta_O) / v;
	tmp = single(0.0);
	if (sinTheta_i <= single(-2.499999990010493e-13))
		tmp = exp(t_0);
	elseif (sinTheta_i <= single(8.00000025099516e-22))
		tmp = t_0;
	else
		tmp = exp((sinTheta_O * (sinTheta_i / -v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if sinTheta_i < -2.49999999e-13

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 25.3%

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

      1. associate-*r/ 25.3%

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

      2. mul-1-neg 25.3%

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

   7. Simplified 25.3%

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

      1. clear-num 25.3%

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

      2. inv-pow 25.3%

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

      3. add-sqr-sqrt 3.3%

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

      4. sqrt-unprod 5.4%

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

      5. *-commutative 5.4%

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

      6. *-commutative 5.4%

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

      7. sqr-neg 5.4%

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

      8. sqrt-unprod 2.0%

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

      9. add-sqr-sqrt 15.4%

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

   9. Applied egg-rr 15.4%

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

       1. unpow-1 15.4%

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

       2. *-commutative 15.4%

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

   11. Simplified 15.4%

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

   12. Taylor expanded in v around 0 15.4%

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

   if -2.49999999e-13 < sinTheta_i < 8.00000025e-22

   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 9.0%

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

      1. associate-*r/ 9.0%

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

      2. mul-1-neg 9.0%

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

   7. Simplified 9.0%

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

      1. clear-num 9.0%

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

      2. inv-pow 9.0%

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

      3. add-sqr-sqrt 4.5%

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

      4. sqrt-unprod 6.4%

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

      5. *-commutative 6.4%

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

      6. *-commutative 6.4%

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

      7. sqr-neg 6.4%

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

      8. sqrt-unprod 4.9%

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

      9. add-sqr-sqrt 7.6%

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

   9. Applied egg-rr 7.6%

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

       1. unpow-1 7.6%

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

       2. *-commutative 7.6%

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

   11. Simplified 7.6%

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

   12. Taylor expanded in v around inf 6.4%

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

   13. Taylor expanded in sinTheta_O around inf 52.8%

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

   if 8.00000025e-22 < sinTheta_i

   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. Step-by-step derivation

      1. associate-+l+ 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/l* 99.6%

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

      4. associate-/l* 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in sinTheta_i around inf 15.8%

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

      1. mul-1-neg 15.8%

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

      2. associate-/l* 15.8%

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

      3. distribute-rgt-neg-in 15.8%

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

      4. distribute-frac-neg2 15.8%

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

   7. Simplified 15.8%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \mathbf{elif}\;sinTheta\_i \leq 8.00000025099516 \cdot 10^{-22}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}\\ \mathbf{elif}\;sinTheta\_i \leq 8.00000025099516 \cdot 10^{-22}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -2.499999990010493e-13)
   (exp (/ 1.0 (/ v (* sinTheta_i sinTheta_O))))
   (if (<= sinTheta_i 8.00000025099516e-22)
     (/ (* sinTheta_i sinTheta_O) v)
     (exp (* sinTheta_O (/ sinTheta_i (- v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= -2.499999990010493e-13f) {
		tmp = expf((1.0f / (v / (sinTheta_i * sinTheta_O))));
	} else if (sinTheta_i <= 8.00000025099516e-22f) {
		tmp = (sinTheta_i * sinTheta_O) / v;
	} else {
		tmp = expf((sinTheta_O * (sinTheta_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 (sintheta_i <= (-2.499999990010493e-13)) then
        tmp = exp((1.0e0 / (v / (sintheta_i * sintheta_o))))
    else if (sintheta_i <= 8.00000025099516e-22) then
        tmp = (sintheta_i * sintheta_o) / v
    else
        tmp = exp((sintheta_o * (sintheta_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 (sinTheta_i <= Float32(-2.499999990010493e-13))
		tmp = exp(Float32(Float32(1.0) / Float32(v / Float32(sinTheta_i * sinTheta_O))));
	elseif (sinTheta_i <= Float32(8.00000025099516e-22))
		tmp = Float32(Float32(sinTheta_i * sinTheta_O) / v);
	else
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / Float32(-v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-2.499999990010493e-13))
		tmp = exp((single(1.0) / (v / (sinTheta_i * sinTheta_O))));
	elseif (sinTheta_i <= single(8.00000025099516e-22))
		tmp = (sinTheta_i * sinTheta_O) / v;
	else
		tmp = exp((sinTheta_O * (sinTheta_i / -v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if sinTheta_i < -2.49999999e-13

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 25.3%

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

      1. associate-*r/ 25.3%

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

      2. mul-1-neg 25.3%

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

   7. Simplified 25.3%

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

      1. clear-num 25.3%

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

      2. inv-pow 25.3%

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

      3. add-sqr-sqrt 3.3%

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

      4. sqrt-unprod 5.4%

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

      5. *-commutative 5.4%

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

      6. *-commutative 5.4%

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

      7. sqr-neg 5.4%

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

      8. sqrt-unprod 2.0%

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

      9. add-sqr-sqrt 15.4%

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

   9. Applied egg-rr 15.4%

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

       1. unpow-1 15.4%

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

       2. *-commutative 15.4%

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

   11. Simplified 15.4%

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

   if -2.49999999e-13 < sinTheta_i < 8.00000025e-22

   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 9.0%

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

      1. associate-*r/ 9.0%

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

      2. mul-1-neg 9.0%

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

   7. Simplified 9.0%

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

      1. clear-num 9.0%

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

      2. inv-pow 9.0%

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

      3. add-sqr-sqrt 4.5%

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

      4. sqrt-unprod 6.4%

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

      5. *-commutative 6.4%

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

      6. *-commutative 6.4%

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

      7. sqr-neg 6.4%

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

      8. sqrt-unprod 4.9%

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

      9. add-sqr-sqrt 7.6%

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

   9. Applied egg-rr 7.6%

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

       1. unpow-1 7.6%

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

       2. *-commutative 7.6%

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

   11. Simplified 7.6%

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

   12. Taylor expanded in v around inf 6.4%

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

   13. Taylor expanded in sinTheta_O around inf 52.8%

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

   if 8.00000025e-22 < sinTheta_i

   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. Step-by-step derivation

      1. associate-+l+ 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/l* 99.6%

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

      4. associate-/l* 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in sinTheta_i around inf 15.8%

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

      1. mul-1-neg 15.8%

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

      2. associate-/l* 15.8%

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

      3. distribute-rgt-neg-in 15.8%

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

      4. distribute-frac-neg2 15.8%

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

   7. Simplified 15.8%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{\frac{1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}\\ \mathbf{elif}\;sinTheta\_i \leq 8.00000025099516 \cdot 10^{-22}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * (1.0e0 / exp(((1.0e0 / v) - 0.6931e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-sum 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.8%

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

   10. +-commutative 99.8%

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

   11. associate--l- 99.8%

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

   12. associate-+r- 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

   1. exp-diff 91.6%

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

   2. clear-num 91.6%

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

   3. +-commutative 91.6%

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

   4. associate-/l* 91.6%

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

   5. fma-define 91.6%

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

7. Applied egg-rr 91.6%

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

8. Taylor expanded in cosTheta_O around 0 99.8%

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

   1. div-exp 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 6: 41.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -2.499999990010493e-13)
   (exp (* sinTheta_O (/ sinTheta_i v)))
   (/ (* sinTheta_i sinTheta_O) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= -2.499999990010493e-13f) {
		tmp = expf((sinTheta_O * (sinTheta_i / v)));
	} else {
		tmp = (sinTheta_i * sinTheta_O) / 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 (sintheta_i <= (-2.499999990010493e-13)) then
        tmp = exp((sintheta_o * (sintheta_i / v)))
    else
        tmp = (sintheta_i * sintheta_o) / v
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-2.499999990010493e-13))
		tmp = exp(Float32(sinTheta_O * Float32(sinTheta_i / v)));
	else
		tmp = Float32(Float32(sinTheta_i * sinTheta_O) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-2.499999990010493e-13))
		tmp = exp((sinTheta_O * (sinTheta_i / v)));
	else
		tmp = (sinTheta_i * sinTheta_O) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -2.49999999e-13

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 25.3%

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

      1. associate-*r/ 25.3%

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

      2. mul-1-neg 25.3%

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

   7. Simplified 25.3%

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

      1. clear-num 25.3%

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

      2. inv-pow 25.3%

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

      3. add-sqr-sqrt 3.3%

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

      4. sqrt-unprod 5.4%

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

      5. *-commutative 5.4%

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

      6. *-commutative 5.4%

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

      7. sqr-neg 5.4%

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

      8. sqrt-unprod 2.0%

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

      9. add-sqr-sqrt 15.4%

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

   9. Applied egg-rr 15.4%

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

       1. unpow-1 15.4%

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

       2. *-commutative 15.4%

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

   11. Simplified 15.4%

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

   12. Taylor expanded in v around 0 15.4%

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

       1. associate-*r/ 15.4%

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

   14. Simplified 15.4%

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

   if -2.49999999e-13 < sinTheta_i

   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 11.5%

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

      1. associate-*r/ 11.5%

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

      2. mul-1-neg 11.5%

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

   7. Simplified 11.5%

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

      1. clear-num 11.5%

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

      2. inv-pow 11.5%

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

      3. add-sqr-sqrt 4.0%

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

      4. sqrt-unprod 6.1%

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

      5. *-commutative 6.1%

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

      6. *-commutative 6.1%

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

      7. sqr-neg 6.1%

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

      8. sqrt-unprod 4.2%

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

      9. add-sqr-sqrt 11.4%

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

   9. Applied egg-rr 11.4%

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

       1. unpow-1 11.4%

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

       2. *-commutative 11.4%

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

   11. Simplified 11.4%

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

   12. Taylor expanded in v around inf 6.3%

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

   13. Taylor expanded in sinTheta_O around inf 40.1%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{sinTheta\_O \cdot \frac{sinTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= sinTheta_i -2.499999990010493e-13)
   (exp (* sinTheta_i (/ sinTheta_O v)))
   (/ (* sinTheta_i sinTheta_O) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (sinTheta_i <= -2.499999990010493e-13f) {
		tmp = expf((sinTheta_i * (sinTheta_O / v)));
	} else {
		tmp = (sinTheta_i * sinTheta_O) / 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 (sintheta_i <= (-2.499999990010493e-13)) then
        tmp = exp((sintheta_i * (sintheta_o / v)))
    else
        tmp = (sintheta_i * sintheta_o) / v
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-2.499999990010493e-13))
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / v)));
	else
		tmp = Float32(Float32(sinTheta_i * sinTheta_O) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (sinTheta_i <= single(-2.499999990010493e-13))
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	else
		tmp = (sinTheta_i * sinTheta_O) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -2.49999999e-13

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 25.3%

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

      1. associate-*r/ 25.3%

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

      2. mul-1-neg 25.3%

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

   7. Simplified 25.3%

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

      1. add-log-exp 25.3%

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

      2. *-un-lft-identity 25.3%

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

      3. log-prod 25.3%

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

      4. metadata-eval 25.3%

         \[\leadsto e^{\color{blue}{0} + \log \left(e^{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}\right)} \]

      5. add-log-exp 25.3%

         \[\leadsto e^{0 + \color{blue}{\frac{-sinTheta\_O \cdot sinTheta\_i}{v}}} \]

      6. add-sqr-sqrt 3.3%

         \[\leadsto e^{0 + \frac{\color{blue}{\sqrt{-sinTheta\_O \cdot sinTheta\_i} \cdot \sqrt{-sinTheta\_O \cdot sinTheta\_i}}}{v}} \]

      7. sqrt-unprod 5.4%

         \[\leadsto e^{0 + \frac{\color{blue}{\sqrt{\left(-sinTheta\_O \cdot sinTheta\_i\right) \cdot \left(-sinTheta\_O \cdot sinTheta\_i\right)}}}{v}} \]

      8. *-commutative 5.4%

         \[\leadsto e^{0 + \frac{\sqrt{\left(-\color{blue}{sinTheta\_i \cdot sinTheta\_O}\right) \cdot \left(-sinTheta\_O \cdot sinTheta\_i\right)}}{v}} \]

      9. *-commutative 5.4%

         \[\leadsto e^{0 + \frac{\sqrt{\left(-sinTheta\_i \cdot sinTheta\_O\right) \cdot \left(-\color{blue}{sinTheta\_i \cdot sinTheta\_O}\right)}}{v}} \]

      10. sqr-neg 5.4%

          \[\leadsto e^{0 + \frac{\sqrt{\color{blue}{\left(sinTheta\_i \cdot sinTheta\_O\right) \cdot \left(sinTheta\_i \cdot sinTheta\_O\right)}}}{v}} \]

      11. sqrt-unprod 2.0%

          \[\leadsto e^{0 + \frac{\color{blue}{\sqrt{sinTheta\_i \cdot sinTheta\_O} \cdot \sqrt{sinTheta\_i \cdot sinTheta\_O}}}{v}} \]

      12. add-sqr-sqrt 15.4%

          \[\leadsto e^{0 + \frac{\color{blue}{sinTheta\_i \cdot sinTheta\_O}}{v}} \]

   9. Applied egg-rr 15.4%

      \[\leadsto e^{\color{blue}{0 + \frac{sinTheta\_i \cdot sinTheta\_O}{v}}} \]
   10. Step-by-step derivation

       1. +-lft-identity 15.4%

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

       2. associate-*r/ 15.4%

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

   11. Simplified 15.4%

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

   if -2.49999999e-13 < sinTheta_i

   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 11.5%

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

      1. associate-*r/ 11.5%

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

      2. mul-1-neg 11.5%

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

   7. Simplified 11.5%

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

      1. clear-num 11.5%

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

      2. inv-pow 11.5%

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

      3. add-sqr-sqrt 4.0%

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

      4. sqrt-unprod 6.1%

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

      5. *-commutative 6.1%

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

      6. *-commutative 6.1%

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

      7. sqr-neg 6.1%

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

      8. sqrt-unprod 4.2%

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

      9. add-sqr-sqrt 11.4%

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

   9. Applied egg-rr 11.4%

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

       1. unpow-1 11.4%

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

       2. *-commutative 11.4%

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

   11. Simplified 11.4%

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

   12. Taylor expanded in v around inf 6.3%

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

   13. Taylor expanded in sinTheta_O around inf 40.1%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* sinTheta_i sinTheta_O) v)))
   (if (<= sinTheta_i -2.499999990010493e-13) (exp t_0) t_0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (sinTheta_i * sinTheta_O) / v;
	float tmp;
	if (sinTheta_i <= -2.499999990010493e-13f) {
		tmp = expf(t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (sintheta_i * sintheta_o) / v
    if (sintheta_i <= (-2.499999990010493e-13)) then
        tmp = exp(t_0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(sinTheta_i * sinTheta_O) / v)
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-2.499999990010493e-13))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (sinTheta_i * sinTheta_O) / v;
	tmp = single(0.0);
	if (sinTheta_i <= single(-2.499999990010493e-13))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -2.49999999e-13

   1. Initial program 100.0%

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

      1. associate-+l+ 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 100.0%

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

      6. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in sinTheta_i around inf 25.3%

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

      1. associate-*r/ 25.3%

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

      2. mul-1-neg 25.3%

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

   7. Simplified 25.3%

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

      1. clear-num 25.3%

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

      2. inv-pow 25.3%

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

      3. add-sqr-sqrt 3.3%

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

      4. sqrt-unprod 5.4%

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

      5. *-commutative 5.4%

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

      6. *-commutative 5.4%

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

      7. sqr-neg 5.4%

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

      8. sqrt-unprod 2.0%

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

      9. add-sqr-sqrt 15.4%

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

   9. Applied egg-rr 15.4%

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

       1. unpow-1 15.4%

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

       2. *-commutative 15.4%

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

   11. Simplified 15.4%

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

   12. Taylor expanded in v around 0 15.4%

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

   if -2.49999999e-13 < sinTheta_i

   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 11.5%

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

      1. associate-*r/ 11.5%

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

      2. mul-1-neg 11.5%

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

   7. Simplified 11.5%

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

      1. clear-num 11.5%

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

      2. inv-pow 11.5%

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

      3. add-sqr-sqrt 4.0%

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

      4. sqrt-unprod 6.1%

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

      5. *-commutative 6.1%

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

      6. *-commutative 6.1%

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

      7. sqr-neg 6.1%

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

      8. sqrt-unprod 4.2%

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

      9. add-sqr-sqrt 11.4%

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

   9. Applied egg-rr 11.4%

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

       1. unpow-1 11.4%

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

       2. *-commutative 11.4%

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

   11. Simplified 11.4%

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

   12. Taylor expanded in v around inf 6.3%

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

   13. Taylor expanded in sinTheta_O around inf 40.1%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.499999990010493 \cdot 10^{-13}:\\ \;\;\;\;e^{\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{sinTheta\_i \cdot sinTheta\_O}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 + ((-1.0e0) / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-sum 99.8%

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

   3. rem-exp-log 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

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

   6. +-rgt-identity 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\color{blue}{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + 0}} \]

   7. metadata-eval 99.8%

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

   8. metadata-eval 99.8%

      \[\leadsto \frac{0.5}{v} \cdot e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \color{blue}{0}} \]

   9. +-rgt-identity 99.8%

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

   10. +-commutative 99.8%

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

   11. associate--l- 99.8%

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

   12. associate-+r- 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in sinTheta_i around 0 99.8%

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

6. Taylor expanded in cosTheta_O around 0 99.8%

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

7. Final simplification 99.8%

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

Alternative 10: 97.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{\frac{-1 + cosTheta\_i \cdot cosTheta\_O}{v}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around 0 99.7%

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

6. Taylor expanded in v around 0 98.1%

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

7. Final simplification 98.1%

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

Alternative 11: 21.1% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ sinTheta\_O \cdot \frac{sinTheta\_i}{v} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
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 13.1%

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

   1. associate-*r/ 13.1%

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

   2. mul-1-neg 13.1%

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

7. Simplified 13.1%

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

   1. clear-num 13.1%

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

   2. inv-pow 13.1%

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

   3. add-sqr-sqrt 3.9%

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

   4. sqrt-unprod 6.0%

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

   5. *-commutative 6.0%

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

   6. *-commutative 6.0%

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

   7. sqr-neg 6.0%

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

   8. sqrt-unprod 4.0%

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

   9. add-sqr-sqrt 11.9%

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

9. Applied egg-rr 11.9%

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

    1. unpow-1 11.9%

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

    2. *-commutative 11.9%

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

11. Simplified 11.9%

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

12. Taylor expanded in v around inf 6.2%

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

13. Taylor expanded in sinTheta_O around inf 36.6%

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

    1. associate-*r/ 18.6%

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

15. Simplified 18.6%

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

16. Final simplification 18.6%

    \[\leadsto sinTheta\_O \cdot \frac{sinTheta\_i}{v} \]
17. Add Preprocessing

Alternative 12: 39.4% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ \frac{sinTheta\_i \cdot sinTheta\_O}{v} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.1%

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

   1. associate-*r/ 13.1%

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

   2. mul-1-neg 13.1%

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

7. Simplified 13.1%

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

   1. clear-num 13.1%

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

   2. inv-pow 13.1%

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

   3. add-sqr-sqrt 3.9%

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

   4. sqrt-unprod 6.0%

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

   5. *-commutative 6.0%

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

   6. *-commutative 6.0%

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

   7. sqr-neg 6.0%

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

   8. sqrt-unprod 4.0%

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

   9. add-sqr-sqrt 11.9%

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

9. Applied egg-rr 11.9%

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

    1. unpow-1 11.9%

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

    2. *-commutative 11.9%

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

11. Simplified 11.9%

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

12. Taylor expanded in v around inf 6.2%

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

13. Taylor expanded in sinTheta_O around inf 36.6%

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

14. Final simplification 36.6%

    \[\leadsto \frac{sinTheta\_i \cdot sinTheta\_O}{v} \]
15. Add Preprocessing

Alternative 13: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l+ 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/l* 99.7%

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

   4. associate-/l* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in sinTheta_i around inf 13.1%

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

   1. associate-*r/ 13.1%

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

   2. mul-1-neg 13.1%

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

7. Simplified 13.1%

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

8. Taylor expanded in sinTheta_O around 0 6.4%

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

9. Final simplification 6.4%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))))