HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 13.7s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.7% 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, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (exp (+ (+ 0.6931 (/ -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 + (-1.0f / v)) + logf(0.5f))) / v;
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in cosTheta_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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   11. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

8. Simplified 99.8%

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

   1. +-commutative N/A

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

   2. log-div N/A

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

   3. associate-+r- N/A

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

   4. exp-diff N/A

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

   5. rem-exp-log N/A

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

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\frac{6931}{10000} + \frac{-1}{v}\right) + \log \frac{1}{2}}\right), \color{blue}{v}\right) \]

   7. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \frac{-1}{v}\right) + \log \frac{1}{2}\right)\right), v\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{6931}{10000} + \frac{-1}{v}\right), \log \frac{1}{2}\right)\right), v\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right), \log \frac{1}{2}\right)\right), v\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right), \log \frac{1}{2}\right)\right), v\right) \]

   11. log-lowering-log.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right), \mathsf{log.f32}\left(\frac{1}{2}\right)\right)\right), v\right) \]

10. Applied egg-rr 99.9%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

5. Simplified 99.8%

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

   1. associate-+l+ N/A

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

   2. log-div N/A

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\log \frac{1}{2}}\right), \color{blue}{\left(e^{\log v - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)}\right)}\right) \]

   6. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\log \frac{1}{2}\right), \left(e^{\color{blue}{\log v - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)}}\right)\right) \]

   7. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \left(e^{\color{blue}{\log v} - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)}\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\left(\log v - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   9. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\log v, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   10. log-lowering-log.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right)\right) \]

   16. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 99.9%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(e^{\left(\log v + \frac{1}{v}\right) - \frac{6931}{10000}}\right)}\right) \]

   2. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\log v + \frac{1}{v}\right) - \frac{6931}{10000}\right)\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\log v + \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\log v, \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\left(\frac{1}{v}\right), \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right) \]

   9. metadata-eval 99.9%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \frac{-6931}{10000}\right)\right)\right)\right) \]

10. Simplified 99.9%

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

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (+ (+ 0.6931 (/ -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 + (-1.0f / v)) + logf((0.5f / v))));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in cosTheta_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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   11. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

5. Simplified 99.8%

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

   1. associate-+l+ N/A

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

   2. log-div N/A

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\log \frac{1}{2}}\right), \color{blue}{\left(e^{\log v - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)}\right)}\right) \]

   6. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\log \frac{1}{2}\right), \left(e^{\color{blue}{\log v - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)}}\right)\right) \]

   7. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \left(e^{\color{blue}{\log v} - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)}\right)\right) \]

   8. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\left(\log v - \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   9. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\log v, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   10. log-lowering-log.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right)\right) \]

   16. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{log.f32}\left(\frac{1}{2}\right)\right), \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right)\right) \]

7. Applied egg-rr 99.9%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(e^{\left(\log v + \frac{1}{v}\right) - \frac{6931}{10000}}\right)}\right) \]

   2. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\log v + \frac{1}{v}\right) - \frac{6931}{10000}\right)\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\log v + \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\log v, \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \left(\frac{1}{v} - \frac{6931}{10000}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \left(\frac{1}{v} + \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\left(\frac{1}{v}\right), \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right) \]

   9. metadata-eval 99.9%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(v\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \frac{-6931}{10000}\right)\right)\right)\right) \]

10. Simplified 99.9%

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

    1. exp-sum N/A

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

    2. rem-exp-log N/A

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

    3. associate-/r* N/A

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

    4. rem-exp-log N/A

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

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(e^{\log \left(\frac{\frac{1}{2}}{v}\right)}\right), \color{blue}{\left(e^{\frac{1}{v} + \frac{-6931}{10000}}\right)}\right) \]

    6. rem-exp-log N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{v}\right), \left(e^{\color{blue}{\frac{1}{v} + \frac{-6931}{10000}}}\right)\right) \]

    7. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \left(e^{\color{blue}{\frac{1}{v} + \frac{-6931}{10000}}}\right)\right) \]

    8. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{exp.f32}\left(\left(\frac{1}{v} + \frac{-6931}{10000}\right)\right)\right) \]

    9. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{v}\right), \frac{-6931}{10000}\right)\right)\right) \]

    10. /-lowering-/.f32 99.8%

        \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \frac{-6931}{10000}\right)\right)\right) \]

12. Applied egg-rr 99.8%

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

Alternative 5: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in v around 0

   \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

   2. *-lowering-*.f32 98.5%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

8. Simplified 98.5%

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

9. Taylor expanded in v around 0

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{\left(\frac{6931}{10000} \cdot v + cosTheta\_O \cdot cosTheta\_i\right) - 1}{v}\right)}\right) \]
10. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\left(\frac{6931}{10000} \cdot v + cosTheta\_O \cdot cosTheta\_i\right) - 1\right), v\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\left(cosTheta\_O \cdot cosTheta\_i + \frac{6931}{10000} \cdot v\right) - 1\right), v\right)\right) \]

    3. associate--l+ N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i + \left(\frac{6931}{10000} \cdot v - 1\right)\right), v\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \left(\frac{6931}{10000} \cdot v - 1\right)\right), v\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \left(\frac{6931}{10000} \cdot v - 1\right)\right), v\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \left(\frac{6931}{10000} \cdot v + \left(\mathsf{neg}\left(1\right)\right)\right)\right), v\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \left(\frac{6931}{10000} \cdot v + -1\right)\right), v\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \left(-1 + \frac{6931}{10000} \cdot v\right)\right), v\right)\right) \]

    9. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \mathsf{+.f32}\left(-1, \left(\frac{6931}{10000} \cdot v\right)\right)\right), v\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \mathsf{+.f32}\left(-1, \left(v \cdot \frac{6931}{10000}\right)\right)\right), v\right)\right) \]

    11. *-lowering-*.f32 98.5%

        \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), \mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(v, \frac{6931}{10000}\right)\right)\right), v\right)\right) \]

11. Simplified 98.5%

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

12. Taylor expanded in cosTheta_O around 0

    \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(\frac{6931}{10000} \cdot v - 1\right)}, v\right)\right) \]
13. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\frac{6931}{10000} \cdot v + \left(\mathsf{neg}\left(1\right)\right)\right), v\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\frac{6931}{10000} \cdot v + -1\right), v\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(-1 + \frac{6931}{10000} \cdot v\right), v\right)\right) \]

    4. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \left(\frac{6931}{10000} \cdot v\right)\right), v\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \left(v \cdot \frac{6931}{10000}\right)\right), v\right)\right) \]

    6. *-lowering-*.f32 98.5%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{*.f32}\left(v, \frac{6931}{10000}\right)\right), v\right)\right) \]

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 6: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in v around 0

   \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}, \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

   2. *-lowering-*.f32 98.5%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

8. Simplified 98.5%

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

9. Taylor expanded in cosTheta_O around 0

   \[\leadsto \color{blue}{e^{\frac{6931}{10000} - \frac{1}{v}}} \]
10. Step-by-step derivation

    1. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(\frac{6931}{10000} - \frac{1}{v}\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right) \]

    3. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right) \]

    4. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right) \]

    6. /-lowering-/.f32 98.5%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right) \]

11. Simplified 98.5%

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

Alternative 7: 98.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[e^{\left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
2. 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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in cosTheta_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. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \log \left(\frac{\frac{1}{2}}{v}\right)\right) - \frac{1}{v}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\log \left(\frac{\frac{1}{2}}{v}\right) + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\log \left(\frac{\frac{1}{2}}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2}}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right) \]

   11. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right) \]

8. Simplified 99.8%

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

9. Taylor expanded in v around 0

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{-1}{v}\right)}\right) \]
10. Step-by-step derivation

    1. /-lowering-/.f32 98.5%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{/.f32}\left(-1, v\right)\right) \]

11. Simplified 98.5%

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

Alternative 8: 42.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\\ t_1 := \frac{v}{sinTheta\_i \cdot sinTheta\_O}\\ \mathbf{if}\;sinTheta\_i \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{0 - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ sinTheta_i (/ v sinTheta_O)))
        (t_1 (/ v (* sinTheta_i sinTheta_O))))
   (if (<= sinTheta_i -2.00000006274879e-22)
     (/ (/ t_0 t_1) (- 0.0 t_0))
     (/ -1.0 t_1))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = sinTheta_i / (v / sinTheta_O);
	float t_1 = v / (sinTheta_i * sinTheta_O);
	float tmp;
	if (sinTheta_i <= -2.00000006274879e-22f) {
		tmp = (t_0 / t_1) / (0.0f - t_0);
	} else {
		tmp = -1.0f / t_1;
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = sintheta_i / (v / sintheta_o)
    t_1 = v / (sintheta_i * sintheta_o)
    if (sintheta_i <= (-2.00000006274879e-22)) then
        tmp = (t_0 / t_1) / (0.0e0 - t_0)
    else
        tmp = (-1.0e0) / t_1
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(sinTheta_i / Float32(v / sinTheta_O))
	t_1 = Float32(v / Float32(sinTheta_i * sinTheta_O))
	tmp = Float32(0.0)
	if (sinTheta_i <= Float32(-2.00000006274879e-22))
		tmp = Float32(Float32(t_0 / t_1) / Float32(Float32(0.0) - t_0));
	else
		tmp = Float32(Float32(-1.0) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = sinTheta_i / (v / sinTheta_O);
	t_1 = v / (sinTheta_i * sinTheta_O);
	tmp = single(0.0);
	if (sinTheta_i <= single(-2.00000006274879e-22))
		tmp = (t_0 / t_1) / (single(0.0) - t_0);
	else
		tmp = single(-1.0) / t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_i < -2.00000006e-22

   1. Initial program 99.7%

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

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 12.5%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 12.5%

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

   6. Taylor expanded in sinTheta_O around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right) \]

      5. *-lowering-*.f32 6.0%

         \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

   8. Simplified 6.0%

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

   9. Taylor expanded in sinTheta_O around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \]

       2. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right) \]

       4. *-lowering-*.f32 21.0%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

   11. Simplified 21.0%

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

       1. neg-sub0 N/A

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

       2. *-commutative N/A

          \[\leadsto 0 - \frac{sinTheta\_i \cdot sinTheta\_O}{v} \]

       3. associate-*r/ N/A

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

       4. flip-- N/A

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

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(0 \cdot 0 - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right), \color{blue}{\left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\left(0 - \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       7. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) \cdot \left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right)\right), \left(\color{blue}{0} + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       8. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) \cdot \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       10. clear-num N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right) \cdot \frac{1}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       11. un-div-inv N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       12. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right), \left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_i \cdot \frac{1}{\frac{v}{sinTheta\_O}}\right), \left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       14. un-div-inv N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}\right), \left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       15. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \left(\frac{v}{sinTheta\_O}\right)\right), \left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       16. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right), \left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       17. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right), \mathsf{/.f32}\left(v, \left(sinTheta\_O \cdot sinTheta\_i\right)\right)\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       18. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right), \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right)\right), \left(0 + sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)\right) \]

       19. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right), \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right)\right), \mathsf{+.f32}\left(0, \color{blue}{\left(sinTheta\_i \cdot \frac{sinTheta\_O}{v}\right)}\right)\right) \]

       20. clear-num N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{/.f32}\left(sinTheta\_i, \mathsf{/.f32}\left(v, sinTheta\_O\right)\right), \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right)\right)\right)\right), \mathsf{+.f32}\left(0, \left(sinTheta\_i \cdot \frac{1}{\color{blue}{\frac{v}{sinTheta\_O}}}\right)\right)\right) \]

   13. Applied egg-rr 30.7%

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

   if -2.00000006e-22 < sinTheta_i

   1. Initial program 99.8%

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

   3. Taylor expanded in sinTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

      5. *-lowering-*.f32 12.4%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

   5. Simplified 12.4%

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

   6. Taylor expanded in sinTheta_O around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right) \]

      5. *-lowering-*.f32 6.3%

         \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

   8. Simplified 6.3%

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

   9. Taylor expanded in sinTheta_O around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \]

       2. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

       3. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right) \]

       4. *-lowering-*.f32 50.7%

          \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

   11. Simplified 50.7%

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

       1. neg-mul-1 N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)}\right) \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(v, \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right)}\right)\right) \]

       6. *-lowering-*.f32 51.0%

          \[\leadsto \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right)\right)\right) \]

   13. Applied egg-rr 51.0%

       \[\leadsto \color{blue}{\frac{-1}{\frac{v}{sinTheta\_O \cdot sinTheta\_i}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{\frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}}{0 - \frac{sinTheta\_i}{\frac{v}{sinTheta\_O}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.2% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}} \end{array} \]

FPCore

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

C

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

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) / (v / (sintheta_i * sintheta_o))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in sinTheta_i around inf

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

   3. --lowering--.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

   5. *-lowering-*.f32 12.4%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

5. Simplified 12.4%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f32 N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right) \]

   5. *-lowering-*.f32 6.2%

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

8. Simplified 6.2%

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

9. Taylor expanded in sinTheta_O around inf

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

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \]

    2. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right) \]

    4. *-lowering-*.f32 43.9%

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

11. Simplified 43.9%

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

    1. neg-mul-1 N/A

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

    2. clear-num N/A

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

    3. un-div-inv N/A

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

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{v}{sinTheta\_O \cdot sinTheta\_i}\right)}\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(v, \color{blue}{\left(sinTheta\_O \cdot sinTheta\_i\right)}\right)\right) \]

    6. *-lowering-*.f32 44.2%

       \[\leadsto \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(v, \mathsf{*.f32}\left(sinTheta\_O, \color{blue}{sinTheta\_i}\right)\right)\right) \]

13. Applied egg-rr 44.2%

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

14. Final simplification 44.2%

    \[\leadsto \frac{-1}{\frac{v}{sinTheta\_i \cdot sinTheta\_O}} \]
15. Add Preprocessing

Alternative 10: 38.9% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{sinTheta\_i \cdot sinTheta\_O}{v} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.0f - ((sinTheta_i * sinTheta_O) / v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.0e0 - ((sintheta_i * sintheta_o) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.0) - Float32(Float32(sinTheta_i * sinTheta_O) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0) - ((sinTheta_i * sinTheta_O) / v);
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in sinTheta_i around inf

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(-1 \cdot \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(0 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

   3. --lowering--.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right)\right) \]

   5. *-lowering-*.f32 12.4%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right)\right) \]

5. Simplified 12.4%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f32 N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), \color{blue}{v}\right)\right) \]

   5. *-lowering-*.f32 6.2%

      \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

8. Simplified 6.2%

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

9. Taylor expanded in sinTheta_O around inf

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

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right) \]

    2. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right) \]

    4. *-lowering-*.f32 43.9%

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right) \]

11. Simplified 43.9%

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

12. Final simplification 43.9%

    \[\leadsto 0 - \frac{sinTheta\_i \cdot sinTheta\_O}{v} \]
13. Add Preprocessing

Alternative 11: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

   3. /-lowering-/.f32 9.5%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

5. Simplified 9.5%

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

6. Taylor expanded in cosTheta_O around 0

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

8. Simplified 6.4%

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

Reproduce

herbie shell --seed 2024158 
(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))))))