HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 17.7s

Alternatives: 11

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) + \frac{6931}{10000}}} \]

   3. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right)} + \frac{6931}{10000}} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   5. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   7. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   8. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right)} \]

   11. mul-1-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right)} \]

   13. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right)} \]

   14. associate-/l* N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right)} \]

   18. +-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right)} \]

5. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   2. lift-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   3. lift-fma.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)} \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   4. lift-/.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) \cdot \color{blue}{\frac{-1}{v}} + \frac{6931}{10000}\right)} \]

   5. lift-fma.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right)}} \]

   6. +-commutative N/A

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   7. lift-log.f32 N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   8. lift-/.f32 N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}} \]

   9. log-div N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \color{blue}{\left(\log \frac{1}{2} - \log v\right)}} \]

   10. associate-+r- N/A

       \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}\right) - \log v}} \]

   11. exp-diff N/A

       \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{e^{\log v}}} \]

   12. rem-exp-log N/A

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{\color{blue}{v}} \]

   13. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{v}} \]

7. Applied rewrites 99.8%

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

   1. lift-fma.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{\mathsf{neg}\left(v\right)}\right) + \log \frac{1}{2}}}{v} \]

   2. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\color{blue}{\mathsf{neg}\left(v\right)}}\right) + \log \frac{1}{2}}}{v} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \color{blue}{\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}}\right) + \log \frac{1}{2}}}{v} \]

   4. lift-+.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right)} + \log \frac{1}{2}}}{v} \]

   5. lift-log.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right) + \color{blue}{\log \frac{1}{2}}}}{v} \]

   6. lift-+.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right) + \log \frac{1}{2}}}}{v} \]

   7. lift-exp.f32 99.8

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

   8. lift-+.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right)} + \log \frac{1}{2}}}{v} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \color{blue}{\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}}\right) + \log \frac{1}{2}}}{v} \]

   10. lift-neg.f32 N/A

       \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\color{blue}{\mathsf{neg}\left(v\right)}}\right) + \log \frac{1}{2}}}{v} \]

   11. distribute-frac-neg2 N/A

       \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)\right)}\right) + \log \frac{1}{2}}}{v} \]

   12. unsub-neg N/A

       \[\leadsto \frac{e^{\color{blue}{\left(\frac{6931}{10000} - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} + \log \frac{1}{2}}}{v} \]

   13. lower--.f32 N/A

       \[\leadsto \frac{e^{\color{blue}{\left(\frac{6931}{10000} - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right)} + \log \frac{1}{2}}}{v} \]

   14. lower-/.f32 99.8

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

9. Applied rewrites 99.8%

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

Alternative 2: 56.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{if}\;\left(0.6931 + \left(\left(t\_0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -999999986991104:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_O cosTheta_i) v)))
   (if (<=
        (+
         (+ 0.6931 (+ (- t_0 (/ (* sinTheta_O sinTheta_i) v)) (/ -1.0 v)))
         (log (/ 1.0 (* v 2.0))))
        -999999986991104.0)
     t_0
     (/
      (*
       (* cosTheta_O (* cosTheta_O cosTheta_O))
       (* 0.16666666666666666 (* cosTheta_i (* cosTheta_i cosTheta_i))))
      (* v (* v v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_O * cosTheta_i) / v;
	float tmp;
	if (((0.6931f + ((t_0 - ((sinTheta_O * sinTheta_i) / v)) + (-1.0f / v))) + logf((1.0f / (v * 2.0f)))) <= -999999986991104.0f) {
		tmp = t_0;
	} else {
		tmp = ((cosTheta_O * (cosTheta_O * cosTheta_O)) * (0.16666666666666666f * (cosTheta_i * (cosTheta_i * cosTheta_i)))) / (v * (v * 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 = (costheta_o * costheta_i) / v
    if (((0.6931e0 + ((t_0 - ((sintheta_o * sintheta_i) / v)) + ((-1.0e0) / v))) + log((1.0e0 / (v * 2.0e0)))) <= (-999999986991104.0e0)) then
        tmp = t_0
    else
        tmp = ((costheta_o * (costheta_o * costheta_o)) * (0.16666666666666666e0 * (costheta_i * (costheta_i * costheta_i)))) / (v * (v * v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_O * cosTheta_i) / v)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(0.6931) + Float32(Float32(t_0 - Float32(Float32(sinTheta_O * sinTheta_i) / v)) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(1.0) / Float32(v * Float32(2.0))))) <= Float32(-999999986991104.0))
		tmp = t_0;
	else
		tmp = Float32(Float32(Float32(cosTheta_O * Float32(cosTheta_O * cosTheta_O)) * Float32(Float32(0.16666666666666666) * Float32(cosTheta_i * Float32(cosTheta_i * cosTheta_i)))) / Float32(v * Float32(v * v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_O * cosTheta_i) / v;
	tmp = single(0.0);
	if (((single(0.6931) + ((t_0 - ((sinTheta_O * sinTheta_i) / v)) + (single(-1.0) / v))) + log((single(1.0) / (v * single(2.0))))) <= single(-999999986991104.0))
		tmp = t_0;
	else
		tmp = ((cosTheta_O * (cosTheta_O * cosTheta_O)) * (single(0.16666666666666666) * (cosTheta_i * (cosTheta_i * cosTheta_i)))) / (v * (v * v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 #s(literal 1 binary32) v)) #s(literal 6931/10000 binary32)) (log.f32 (/.f32 #s(literal 1 binary32) (*.f32 #s(literal 2 binary32) v)))) < -9.99999987e14

   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. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 15.9

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

   5. Applied rewrites 15.9%

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

   6. Taylor expanded in cosTheta_i around 0

      \[\leadsto \color{blue}{1 + cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right) + 1} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}, 1\right)} \]

   8. Applied rewrites 0.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(0.5, \frac{cosTheta\_O \cdot cosTheta\_O}{v \cdot v}, \frac{\left(0.16666666666666666 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right) \cdot cosTheta\_i}{v \cdot \left(v \cdot v\right)}\right), \frac{cosTheta\_O}{v}\right), 1\right)} \]

   9. Taylor expanded in cosTheta_i around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-/.f32 6.0

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

   11. Applied rewrites 6.0%

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

   12. Taylor expanded in cosTheta_O around inf

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

       1. lower-/.f32 N/A

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

       2. lower-*.f32 38.2

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

   14. Applied rewrites 38.2%

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

   if -9.99999987e14 < (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 #s(literal 1 binary32) v)) #s(literal 6931/10000 binary32)) (log.f32 (/.f32 #s(literal 1 binary32) (*.f32 #s(literal 2 binary32) v))))

   1. Initial program 98.9%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in cosTheta_i around 0

      \[\leadsto \color{blue}{1 + cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right) + 1} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}, 1\right)} \]

   8. Applied rewrites 6.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(0.5, \frac{cosTheta\_O \cdot cosTheta\_O}{v \cdot v}, \frac{\left(0.16666666666666666 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right) \cdot cosTheta\_i}{v \cdot \left(v \cdot v\right)}\right), \frac{cosTheta\_O}{v}\right), 1\right)} \]

   9. Taylor expanded in cosTheta_i around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot {cosTheta\_i}^{3}}{{v}^{3}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot \left({cosTheta\_O}^{3} \cdot {cosTheta\_i}^{3}\right)}{{v}^{3}}} \]

       2. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{6} \cdot \left({cosTheta\_O}^{3} \cdot {cosTheta\_i}^{3}\right)}{{v}^{3}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({cosTheta\_O}^{3} \cdot {cosTheta\_i}^{3}\right) \cdot \frac{1}{6}}}{{v}^{3}} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{cosTheta\_O}^{3} \cdot \left({cosTheta\_i}^{3} \cdot \frac{1}{6}\right)}}{{v}^{3}} \]

       5. *-commutative N/A

          \[\leadsto \frac{{cosTheta\_O}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}}{{v}^{3}} \]

       6. lower-*.f32 N/A

          \[\leadsto \frac{\color{blue}{{cosTheta\_O}^{3} \cdot \left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}}{{v}^{3}} \]

       7. cube-mult N/A

          \[\leadsto \frac{\color{blue}{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)} \cdot \left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}{{v}^{3}} \]

       8. unpow2 N/A

          \[\leadsto \frac{\left(cosTheta\_O \cdot \color{blue}{{cosTheta\_O}^{2}}\right) \cdot \left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}{{v}^{3}} \]

       9. lower-*.f32 N/A

          \[\leadsto \frac{\color{blue}{\left(cosTheta\_O \cdot {cosTheta\_O}^{2}\right)} \cdot \left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}{{v}^{3}} \]

       10. unpow2 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_O\right)}\right) \cdot \left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}{{v}^{3}} \]

       11. lower-*.f32 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_O\right)}\right) \cdot \left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}{{v}^{3}} \]

       12. lower-*.f32 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {cosTheta\_i}^{3}\right)}}{{v}^{3}} \]

       13. cube-mult N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)}\right)}{{v}^{3}} \]

       14. unpow2 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot \color{blue}{{cosTheta\_i}^{2}}\right)\right)}{{v}^{3}} \]

       15. lower-*.f32 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(cosTheta\_i \cdot {cosTheta\_i}^{2}\right)}\right)}{{v}^{3}} \]

       16. unpow2 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)}\right)\right)}{{v}^{3}} \]

       17. lower-*.f32 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)}\right)\right)}{{v}^{3}} \]

       18. cube-mult N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{\color{blue}{v \cdot \left(v \cdot v\right)}} \]

       19. unpow2 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot \color{blue}{{v}^{2}}} \]

       20. lower-*.f32 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{\color{blue}{v \cdot {v}^{2}}} \]

       21. unpow2 N/A

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot \color{blue}{\left(v \cdot v\right)}} \]

       22. lower-*.f32 82.8

           \[\leadsto \frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot \color{blue}{\left(v \cdot v\right)}} \]

   11. Applied rewrites 82.8%

       \[\leadsto \color{blue}{\frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot \left(v \cdot v\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.6931 + \left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -999999986991104:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)\right)}{v \cdot \left(v \cdot v\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{if}\;\left(0.6931 + \left(\left(t\_0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -999999986991104:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{cosTheta\_i \cdot \left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)}{v \cdot \left(v \cdot v\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_O cosTheta_i) v)))
   (if (<=
        (+
         (+ 0.6931 (+ (- t_0 (/ (* sinTheta_O sinTheta_i) v)) (/ -1.0 v)))
         (log (/ 1.0 (* v 2.0))))
        -999999986991104.0)
     t_0
     (*
      (* cosTheta_O (* cosTheta_O cosTheta_O))
      (/
       (* cosTheta_i (* 0.16666666666666666 (* cosTheta_i cosTheta_i)))
       (* v (* v v)))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_O * cosTheta_i) / v;
	float tmp;
	if (((0.6931f + ((t_0 - ((sinTheta_O * sinTheta_i) / v)) + (-1.0f / v))) + logf((1.0f / (v * 2.0f)))) <= -999999986991104.0f) {
		tmp = t_0;
	} else {
		tmp = (cosTheta_O * (cosTheta_O * cosTheta_O)) * ((cosTheta_i * (0.16666666666666666f * (cosTheta_i * cosTheta_i))) / (v * (v * 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 = (costheta_o * costheta_i) / v
    if (((0.6931e0 + ((t_0 - ((sintheta_o * sintheta_i) / v)) + ((-1.0e0) / v))) + log((1.0e0 / (v * 2.0e0)))) <= (-999999986991104.0e0)) then
        tmp = t_0
    else
        tmp = (costheta_o * (costheta_o * costheta_o)) * ((costheta_i * (0.16666666666666666e0 * (costheta_i * costheta_i))) / (v * (v * v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_O * cosTheta_i) / v)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(0.6931) + Float32(Float32(t_0 - Float32(Float32(sinTheta_O * sinTheta_i) / v)) + Float32(Float32(-1.0) / v))) + log(Float32(Float32(1.0) / Float32(v * Float32(2.0))))) <= Float32(-999999986991104.0))
		tmp = t_0;
	else
		tmp = Float32(Float32(cosTheta_O * Float32(cosTheta_O * cosTheta_O)) * Float32(Float32(cosTheta_i * Float32(Float32(0.16666666666666666) * Float32(cosTheta_i * cosTheta_i))) / Float32(v * Float32(v * v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_O * cosTheta_i) / v;
	tmp = single(0.0);
	if (((single(0.6931) + ((t_0 - ((sinTheta_O * sinTheta_i) / v)) + (single(-1.0) / v))) + log((single(1.0) / (v * single(2.0))))) <= single(-999999986991104.0))
		tmp = t_0;
	else
		tmp = (cosTheta_O * (cosTheta_O * cosTheta_O)) * ((cosTheta_i * (single(0.16666666666666666) * (cosTheta_i * cosTheta_i))) / (v * (v * v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 #s(literal 1 binary32) v)) #s(literal 6931/10000 binary32)) (log.f32 (/.f32 #s(literal 1 binary32) (*.f32 #s(literal 2 binary32) v)))) < -9.99999987e14

   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. Add Preprocessing

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 15.9

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

   5. Applied rewrites 15.9%

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

   6. Taylor expanded in cosTheta_i around 0

      \[\leadsto \color{blue}{1 + cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right) + 1} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}, 1\right)} \]

   8. Applied rewrites 0.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(0.5, \frac{cosTheta\_O \cdot cosTheta\_O}{v \cdot v}, \frac{\left(0.16666666666666666 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right) \cdot cosTheta\_i}{v \cdot \left(v \cdot v\right)}\right), \frac{cosTheta\_O}{v}\right), 1\right)} \]

   9. Taylor expanded in cosTheta_i around 0

      \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-/.f32 6.0

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

   11. Applied rewrites 6.0%

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

   12. Taylor expanded in cosTheta_O around inf

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

       1. lower-/.f32 N/A

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

       2. lower-*.f32 38.2

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

   14. Applied rewrites 38.2%

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

   if -9.99999987e14 < (+.f32 (+.f32 (-.f32 (-.f32 (/.f32 (*.f32 cosTheta_i cosTheta_O) v) (/.f32 (*.f32 sinTheta_i sinTheta_O) v)) (/.f32 #s(literal 1 binary32) v)) #s(literal 6931/10000 binary32)) (log.f32 (/.f32 #s(literal 1 binary32) (*.f32 #s(literal 2 binary32) v))))

   1. Initial program 98.9%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in cosTheta_i around 0

      \[\leadsto \color{blue}{1 + cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right) + 1} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}, 1\right)} \]

   8. Applied rewrites 6.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(0.5, \frac{cosTheta\_O \cdot cosTheta\_O}{v \cdot v}, \frac{\left(0.16666666666666666 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right) \cdot cosTheta\_i}{v \cdot \left(v \cdot v\right)}\right), \frac{cosTheta\_O}{v}\right), 1\right)} \]

   10. Applied rewrites 6.8%

       \[\leadsto \color{blue}{cosTheta\_i \cdot \mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right), 0.16666666666666666 \cdot \frac{cosTheta\_i}{v \cdot \left(v \cdot v\right)}, \frac{0.5 \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)}{v \cdot v}\right), \frac{cosTheta\_O}{v}\right) + 1} \]

   11. Taylor expanded in cosTheta_i around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot {cosTheta\_i}^{3}}{{v}^{3}}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{{cosTheta\_O}^{3} \cdot {cosTheta\_i}^{3}}{{v}^{3}} \cdot \frac{1}{6}} \]

       2. associate-/l* N/A

          \[\leadsto \color{blue}{\left({cosTheta\_O}^{3} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right)} \cdot \frac{1}{6} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{{cosTheta\_O}^{3} \cdot \left(\frac{{cosTheta\_i}^{3}}{{v}^{3}} \cdot \frac{1}{6}\right)} \]

       4. *-commutative N/A

          \[\leadsto {cosTheta\_O}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right)} \]

       5. lower-*.f32 N/A

          \[\leadsto \color{blue}{{cosTheta\_O}^{3} \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right)} \]

       6. cube-mult N/A

          \[\leadsto \color{blue}{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)} \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right) \]

       7. unpow2 N/A

          \[\leadsto \left(cosTheta\_O \cdot \color{blue}{{cosTheta\_O}^{2}}\right) \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right) \]

       8. lower-*.f32 N/A

          \[\leadsto \color{blue}{\left(cosTheta\_O \cdot {cosTheta\_O}^{2}\right)} \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right) \]

       9. unpow2 N/A

          \[\leadsto \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_O\right)}\right) \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right) \]

       10. lower-*.f32 N/A

           \[\leadsto \left(cosTheta\_O \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_O\right)}\right) \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_i}^{3}}{{v}^{3}}\right) \]

       11. associate-*r/ N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \color{blue}{\frac{\frac{1}{6} \cdot {cosTheta\_i}^{3}}{{v}^{3}}} \]

       12. lower-/.f32 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \color{blue}{\frac{\frac{1}{6} \cdot {cosTheta\_i}^{3}}{{v}^{3}}} \]

       13. unpow3 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\frac{1}{6} \cdot \color{blue}{\left(\left(cosTheta\_i \cdot cosTheta\_i\right) \cdot cosTheta\_i\right)}}{{v}^{3}} \]

       14. unpow2 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\frac{1}{6} \cdot \left(\color{blue}{{cosTheta\_i}^{2}} \cdot cosTheta\_i\right)}{{v}^{3}} \]

       15. associate-*r* N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot {cosTheta\_i}^{2}\right) \cdot cosTheta\_i}}{{v}^{3}} \]

       16. lower-*.f32 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot {cosTheta\_i}^{2}\right) \cdot cosTheta\_i}}{{v}^{3}} \]

       17. lower-*.f32 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\color{blue}{\left(\frac{1}{6} \cdot {cosTheta\_i}^{2}\right)} \cdot cosTheta\_i}{{v}^{3}} \]

       18. unpow2 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(\frac{1}{6} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)}\right) \cdot cosTheta\_i}{{v}^{3}} \]

       19. lower-*.f32 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(\frac{1}{6} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_i\right)}\right) \cdot cosTheta\_i}{{v}^{3}} \]

       20. cube-mult N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot cosTheta\_i}{\color{blue}{v \cdot \left(v \cdot v\right)}} \]

       21. unpow2 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot cosTheta\_i}{v \cdot \color{blue}{{v}^{2}}} \]

       22. lower-*.f32 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot cosTheta\_i}{\color{blue}{v \cdot {v}^{2}}} \]

       23. unpow2 N/A

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(\frac{1}{6} \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot cosTheta\_i}{v \cdot \color{blue}{\left(v \cdot v\right)}} \]

       24. lower-*.f32 80.7

           \[\leadsto \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot cosTheta\_i}{v \cdot \color{blue}{\left(v \cdot v\right)}} \]

   13. Applied rewrites 80.7%

       \[\leadsto \color{blue}{\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{\left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right) \cdot cosTheta\_i}{v \cdot \left(v \cdot v\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.6931 + \left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{-1}{v}\right)\right) + \log \left(\frac{1}{v \cdot 2}\right) \leq -999999986991104:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right) \cdot \frac{cosTheta\_i \cdot \left(0.16666666666666666 \cdot \left(cosTheta\_i \cdot cosTheta\_i\right)\right)}{v \cdot \left(v \cdot v\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) + \frac{6931}{10000}}} \]

   3. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right)} + \frac{6931}{10000}} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   5. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   7. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   8. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right)} \]

   11. mul-1-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right)} \]

   13. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right)} \]

   14. associate-/l* N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right)} \]

   18. +-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right)} \]

5. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   2. lift-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   3. lift-fma.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)} \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   4. lift-/.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) \cdot \color{blue}{\frac{-1}{v}} + \frac{6931}{10000}\right)} \]

   5. lift-fma.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right)}} \]

   6. +-commutative N/A

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   7. lift-log.f32 N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   8. lift-/.f32 N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}} \]

   9. log-div N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \color{blue}{\left(\log \frac{1}{2} - \log v\right)}} \]

   10. associate-+r- N/A

       \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}\right) - \log v}} \]

   11. exp-diff N/A

       \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{e^{\log v}}} \]

   12. rem-exp-log N/A

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{\color{blue}{v}} \]

   13. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{v}} \]

7. Applied rewrites 99.8%

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

   1. lift-fma.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \frac{\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}}{\mathsf{neg}\left(v\right)}\right) + \log \frac{1}{2}}}{v} \]

   2. lift-neg.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\color{blue}{\mathsf{neg}\left(v\right)}}\right) + \log \frac{1}{2}}}{v} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \color{blue}{\frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}}\right) + \log \frac{1}{2}}}{v} \]

   4. lift-+.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right)} + \log \frac{1}{2}}}{v} \]

   5. lift-log.f32 N/A

      \[\leadsto \frac{e^{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right) + \color{blue}{\log \frac{1}{2}}}}{v} \]

   6. exp-sum N/A

      \[\leadsto \frac{\color{blue}{e^{\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}} \cdot e^{\log \frac{1}{2}}}}{v} \]

   7. lift-log.f32 N/A

      \[\leadsto \frac{e^{\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}} \cdot e^{\color{blue}{\log \frac{1}{2}}}}{v} \]

   8. rem-exp-log N/A

      \[\leadsto \frac{e^{\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}} \cdot \color{blue}{\frac{1}{2}}}{v} \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}} \cdot \frac{\frac{1}{2}}{v}} \]

   10. lift-/.f32 N/A

       \[\leadsto e^{\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}} \cdot \color{blue}{\frac{\frac{1}{2}}{v}} \]

   11. rem-exp-log N/A

       \[\leadsto e^{\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}} \cdot \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   12. lift-log.f32 N/A

       \[\leadsto e^{\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}} \cdot e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   13. prod-exp N/A

       \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   14. lower-exp.f32 N/A

       \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{\mathsf{neg}\left(v\right)}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

    \[\leadsto e^{\left(0.6931 - \frac{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)}{v}\right) - \log \left(v \cdot 2\right)} \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (- 0.6931 (/ (fma sinTheta_O sinTheta_i 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 - (fmaf(sinTheta_O, sinTheta_i, 1.0f) / v)));
}

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(fma(sinTheta_O, sinTheta_i, Float32(1.0)) / v))))
end

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) + \frac{6931}{10000}}} \]

   3. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right)} + \frac{6931}{10000}} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   5. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   7. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   8. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right)} \]

   11. mul-1-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right)} \]

   13. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right)} \]

   14. associate-/l* N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right)} \]

   18. +-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right)} \]

5. Applied rewrites 99.7%

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

6. Taylor expanded in v around 0

   \[\leadsto \color{blue}{e^{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}\right) + \frac{6931}{10000}}} \]

   2. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(-1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v} + \frac{6931}{10000}\right)}} \]

   3. +-commutative N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]

   4. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   5. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}} \]

   8. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} + -1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\left(\mathsf{neg}\left(\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   10. unsub-neg N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   11. lower--.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} - \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   12. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \color{blue}{\frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}}} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{\color{blue}{sinTheta\_O \cdot sinTheta\_i + 1}}{v}} \]

   14. lower-fma.f32 99.7

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

8. Applied rewrites 99.7%

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

Alternative 6: 99.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) + \frac{6931}{10000}}} \]

   3. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right)} + \frac{6931}{10000}} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   5. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   7. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   8. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right)} \]

   11. mul-1-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right)} \]

   13. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right)} \]

   14. associate-/l* N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right)} \]

   18. +-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right)} \]

5. Applied rewrites 99.7%

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

6. Taylor expanded in sinTheta_O around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \frac{1}{v}}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right)} - \frac{1}{v}} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)}} \]

   3. exp-sum N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{1}{2}}{v}\right)} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}} \]

   4. rem-exp-log N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \frac{1}{v}} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} - \frac{1}{v}}} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{v}} \cdot e^{\frac{6931}{10000} - \frac{1}{v}} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot \color{blue}{e^{\frac{6931}{10000} - \frac{1}{v}}} \]

   8. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

   9. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\color{blue}{\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}} \]

   10. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{v} \cdot e^{\frac{6931}{10000} + \frac{\color{blue}{-1}}{v}} \]

   12. lower-/.f32 99.5

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

8. Applied rewrites 99.5%

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

Alternative 7: 99.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) + \frac{6931}{10000}}} \]

   3. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right)} + \frac{6931}{10000}} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   5. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   7. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   8. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right)} \]

   11. mul-1-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right)} \]

   13. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right)} \]

   14. associate-/l* N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right)} \]

   18. +-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right)} \]

5. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   2. lift-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   3. lift-fma.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right)} \cdot \frac{-1}{v} + \frac{6931}{10000}\right)} \]

   4. lift-/.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right) \cdot \color{blue}{\frac{-1}{v}} + \frac{6931}{10000}\right)} \]

   5. lift-fma.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right)}} \]

   6. +-commutative N/A

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   7. lift-log.f32 N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right)}} \]

   8. lift-/.f32 N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)}} \]

   9. log-div N/A

      \[\leadsto e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \color{blue}{\left(\log \frac{1}{2} - \log v\right)}} \]

   10. associate-+r- N/A

       \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}\right) - \log v}} \]

   11. exp-diff N/A

       \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{e^{\log v}}} \]

   12. rem-exp-log N/A

       \[\leadsto \frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{\color{blue}{v}} \]

   13. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_i, 1\right), \frac{-1}{v}, \frac{6931}{10000}\right) + \log \frac{1}{2}}}{v}} \]

7. Applied rewrites 99.8%

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

8. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 99.1

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

10. Applied rewrites 99.1%

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

11. Taylor expanded in sinTheta_O around 0

    \[\leadsto \color{blue}{\frac{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}{v}} \]
12. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}{v}} \]

    2. lower-exp.f32 N/A

       \[\leadsto \frac{\color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}}{v} \]

    3. distribute-neg-frac N/A

       \[\leadsto \frac{e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}}{v} \]

    4. metadata-eval N/A

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

    5. lower-/.f32 99.0

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

13. Applied rewrites 99.0%

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

Alternative 8: 97.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]
4. Step-by-step derivation

   1. lower-exp.f32 N/A

      \[\leadsto \color{blue}{e^{\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}}} \]

   2. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)}} \]

   3. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)}} \]

   4. associate--l+ N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   5. lower-+.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)}} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   7. lower-log.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   8. rem-exp-log N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \left(\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)} \]

   10. div-sub N/A

       \[\leadsto e^{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   11. lower-/.f32 N/A

       \[\leadsto e^{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}\right)} \]

   12. sub-neg N/A

       \[\leadsto e^{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}\right)} \]

   13. metadata-eval N/A

       \[\leadsto e^{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}\right)} \]

   14. lower-fma.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in v around 0

   \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto e^{\frac{6931}{10000} + \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - 1}{v}}} \]

   2. sub-neg N/A

      \[\leadsto e^{\frac{6931}{10000} + \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i + \left(\mathsf{neg}\left(1\right)\right)}}{v}} \]

   3. metadata-eval N/A

      \[\leadsto e^{\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i + \color{blue}{-1}}{v}} \]

   4. lower-fma.f32 97.5

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

8. Applied rewrites 97.5%

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

Alternative 9: 97.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around 0

   \[\leadsto e^{\color{blue}{\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto e^{\color{blue}{\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) - \left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) + \frac{6931}{10000}}} \]

   3. sub-neg N/A

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)\right)} + \frac{6931}{10000}} \]

   4. associate-+l+ N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   5. lower-+.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)}} \]

   6. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   7. lower-log.f32 N/A

      \[\leadsto e^{\color{blue}{\log \left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   8. rem-exp-log N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   9. lower-/.f32 N/A

      \[\leadsto e^{\log \color{blue}{\left(\frac{\frac{1}{2}}{v}\right)} + \left(\left(\mathsf{neg}\left(\left(\frac{1}{v} + \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) + \frac{6931}{10000}\right)} \]

   10. distribute-neg-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right)} + \frac{6931}{10000}\right)} \]

   11. mul-1-neg N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}}\right) + \frac{6931}{10000}\right)} \]

   12. associate-*r/ N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\frac{-1 \cdot \left(sinTheta\_O \cdot sinTheta\_i\right)}{v}}\right) + \frac{6931}{10000}\right)} \]

   13. *-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot -1}}{v}\right) + \frac{6931}{10000}\right)} \]

   14. associate-/l* N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{-1}{v}}\right) + \frac{6931}{10000}\right)} \]

   15. metadata-eval N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}\right) + \frac{6931}{10000}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\left(\left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \left(sinTheta\_O \cdot sinTheta\_i\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)}\right) + \frac{6931}{10000}\right)} \]

   17. distribute-rgt1-in N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_i + 1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} + \frac{6931}{10000}\right)} \]

   18. +-commutative N/A

       \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\color{blue}{\left(1 + sinTheta\_O \cdot sinTheta\_i\right)} \cdot \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) + \frac{6931}{10000}\right)} \]

5. Applied rewrites 99.7%

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

6. Taylor expanded in sinTheta_O around 0

   \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} - \frac{1}{v}\right)}} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)}} \]

   2. lower-+.f32 N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \color{blue}{\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)}} \]

   3. distribute-neg-frac N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}\right)} \]

   4. metadata-eval N/A

      \[\leadsto e^{\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} + \frac{\color{blue}{-1}}{v}\right)} \]

   5. lower-/.f32 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{-1}{v}}} \]
10. Step-by-step derivation

    1. lower-/.f32 97.5

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

11. Applied rewrites 97.5%

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

Alternative 10: 39.2% accurate, 16.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 13.2

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

5. Applied rewrites 13.2%

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

6. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{1 + cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{cosTheta\_i \cdot \left(cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}\right) + 1} \]

   2. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, cosTheta\_i \cdot \left(\frac{1}{6} \cdot \frac{{cosTheta\_O}^{3} \cdot cosTheta\_i}{{v}^{3}} + \frac{1}{2} \cdot \frac{{cosTheta\_O}^{2}}{{v}^{2}}\right) + \frac{cosTheta\_O}{v}, 1\right)} \]

8. Applied rewrites 2.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(cosTheta\_i, \mathsf{fma}\left(0.5, \frac{cosTheta\_O \cdot cosTheta\_O}{v \cdot v}, \frac{\left(0.16666666666666666 \cdot \left(cosTheta\_O \cdot \left(cosTheta\_O \cdot cosTheta\_O\right)\right)\right) \cdot cosTheta\_i}{v \cdot \left(v \cdot v\right)}\right), \frac{cosTheta\_O}{v}\right), 1\right)} \]

9. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. associate-/l* N/A

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

    3. lower-fma.f32 N/A

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

    4. lower-/.f32 6.3

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

11. Applied rewrites 6.3%

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

12. Taylor expanded in cosTheta_O around inf

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

    1. lower-/.f32 N/A

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

    2. lower-*.f32 38.5

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

14. Applied rewrites 38.5%

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

Alternative 11: 6.4% accurate, 272.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 13.2

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

5. Applied rewrites 13.2%

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

6. Taylor expanded in cosTheta_i around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 6.5%

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

Reproduce

herbie shell --seed 2024214 
(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))))))