HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.7%

Time: 15.5s

Alternatives: 9

Speedup: 1.9×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

   1. div-inv N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. prod-exp N/A

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

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

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

   6. rem-log-exp N/A

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

   7. pow-to-exp N/A

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

   8. inv-pow N/A

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

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

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{\left(cosTheta\_O \cdot cosTheta\_i + v \cdot \left(\frac{6931}{10000} - \log v\right)\right) - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right)\right) \]
8. Step-by-step derivation

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

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

   2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-lowering-*.f32 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in cosTheta_O around 0

    \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{v \cdot \left(\frac{6931}{10000} - \log v\right) - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right)\right) \]
11. Step-by-step derivation

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

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

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

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

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

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

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

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

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

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

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

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

    7. *-lowering-*.f32 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((((v * (0.6931e0 - log(v))) + ((-1.0e0) - (sintheta_o * sintheta_i))) / v)) * 0.5e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

   1. div-inv N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. prod-exp N/A

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

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

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

   6. rem-log-exp N/A

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

   7. pow-to-exp N/A

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

   8. inv-pow N/A

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

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

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(2, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{\left(cosTheta\_O \cdot cosTheta\_i + v \cdot \left(\frac{6931}{10000} - \log v\right)\right) - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)}\right)\right)\right) \]
8. Step-by-step derivation

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

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

   2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-lowering-*.f32 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in cosTheta_O around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    9. *-lowering-*.f32 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. log-rec N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

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

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

3. Simplified 99.7%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. div-inv N/A

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

   4. neg-mul-1 N/A

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

   5. sub-neg N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. associate-+l- N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. exp-diff N/A

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

6. Applied egg-rr 87.6%

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

7. Taylor expanded in v around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(\left(1 + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, v\right), \frac{-6931}{10000}\right)\right)\right), \mathsf{*.f32}\left(v, 2\right)\right) \]
8. Step-by-step derivation

   1. associate--l+ N/A

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

   2. sub-neg N/A

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

   3. mul-1-neg N/A

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. sub-neg N/A

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

   9. div-sub N/A

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

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

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

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

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

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

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

   13. *-lowering-*.f32 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in cosTheta_O around 0

    \[\leadsto \mathsf{/.f32}\left(\color{blue}{\left(\frac{1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{e^{\frac{1}{v} - \frac{6931}{10000}}}\right)}, \mathsf{*.f32}\left(v, 2\right)\right) \]
11. Step-by-step derivation

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

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

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \left(\frac{sinTheta\_O \cdot sinTheta\_i}{v}\right)\right), \left(e^{\frac{1}{v} - \frac{6931}{10000}}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(sinTheta\_O \cdot sinTheta\_i\right), v\right)\right), \left(e^{\frac{1}{v} - \frac{6931}{10000}}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(sinTheta\_O, sinTheta\_i\right), v\right)\right), \left(e^{\frac{1}{v} - \frac{6931}{10000}}\right)\right), \mathsf{*.f32}\left(v, 2\right)\right) \]

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

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

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

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

    9. /-lowering-/.f32 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

    \[\leadsto \frac{\frac{1 - \frac{sinTheta\_O \cdot sinTheta\_i}{v}}{e^{\frac{1}{v} + -0.6931}}}{2 \cdot v} \]
14. Add Preprocessing

Alternative 4: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 4.19999999405965 \cdot 10^{-13}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{cosTheta\_O}{\frac{v}{cosTheta\_i}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 4.19999999405965e-13)
   (/ (* cosTheta_O cosTheta_i) v)
   (exp (/ cosTheta_O (/ v cosTheta_i)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_O <= 4.19999999405965e-13f) {
		tmp = (cosTheta_O * cosTheta_i) / v;
	} else {
		tmp = expf((cosTheta_O / (v / cosTheta_i)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (costheta_o <= 4.19999999405965e-13) then
        tmp = (costheta_o * costheta_i) / v
    else
        tmp = exp((costheta_o / (v / costheta_i)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_O <= Float32(4.19999999405965e-13))
		tmp = Float32(Float32(cosTheta_O * cosTheta_i) / v);
	else
		tmp = exp(Float32(cosTheta_O / Float32(v / cosTheta_i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (cosTheta_O <= single(4.19999999405965e-13))
		tmp = (cosTheta_O * cosTheta_i) / v;
	else
		tmp = exp((cosTheta_O / (v / cosTheta_i)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 4.19999999e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   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 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in cosTheta_O around 0

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

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

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

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

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

      3. *-lowering-*.f32 6.3%

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

   8. Simplified 6.3%

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

   9. Taylor expanded in cosTheta_O around inf

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

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

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

       2. *-lowering-*.f32 45.1%

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

   11. Simplified 45.1%

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

   if 4.19999999e-13 < cosTheta_O

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   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 30.1%

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

   5. Simplified 30.1%

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

      1. clear-num N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{1}{\frac{v}{cosTheta\_i}}\right)\right) \]

      2. un-div-inv N/A

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

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

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

      4. /-lowering-/.f32 30.1%

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

   7. Applied egg-rr 30.1%

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

Alternative 5: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 4.19999999405965 \cdot 10^{-13}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 4.19999999405965e-13)
   (/ (* cosTheta_O cosTheta_i) v)
   (exp (* cosTheta_O (/ cosTheta_i v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_O <= 4.19999999405965e-13f) {
		tmp = (cosTheta_O * cosTheta_i) / v;
	} else {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (costheta_o <= 4.19999999405965e-13) then
        tmp = (costheta_o * costheta_i) / v
    else
        tmp = exp((costheta_o * (costheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_O <= Float32(4.19999999405965e-13))
		tmp = Float32(Float32(cosTheta_O * cosTheta_i) / v);
	else
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (cosTheta_O <= single(4.19999999405965e-13))
		tmp = (cosTheta_O * cosTheta_i) / v;
	else
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 4.19999999e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   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 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in cosTheta_O around 0

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

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

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

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

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

      3. *-lowering-*.f32 6.3%

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

   8. Simplified 6.3%

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

   9. Taylor expanded in cosTheta_O around inf

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

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

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

       2. *-lowering-*.f32 45.1%

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

   11. Simplified 45.1%

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

   if 4.19999999e-13 < cosTheta_O

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   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 30.1%

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

   5. Simplified 30.1%

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

Alternative 6: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}, 1\right), v\right), \frac{6931}{10000}\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, v\right), \color{blue}{\left(e^{\frac{6931}{10000} - \frac{1}{v}}\right)}\right) \]
9. Step-by-step derivation

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

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

   2. sub-neg N/A

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

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

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f32 99.7%

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

10. Simplified 99.7%

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

Alternative 7: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

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

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

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

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

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

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

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

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

   5. *-lowering-*.f32 97.8%

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

5. Simplified 97.8%

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

6. Taylor expanded in cosTheta_O around 0

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

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

      \[\leadsto \mathsf{exp.f32}\left(\left(-1 \cdot \frac{1 + sinTheta\_O \cdot sinTheta\_i}{v}\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\frac{-1 \cdot \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\frac{\mathsf{neg}\left(\left(1 + sinTheta\_O \cdot sinTheta\_i\right)\right)}{v}\right)\right) \]

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

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

   5. distribute-neg-in N/A

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

   6. metadata-eval N/A

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

   7. unsub-neg N/A

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

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

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

   9. *-lowering-*.f32 97.8%

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

8. Simplified 97.8%

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

Alternative 8: 38.0% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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 13.5%

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

5. Simplified 13.5%

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

6. Taylor expanded in cosTheta_O around 0

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

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

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

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

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

   3. *-lowering-*.f32 6.3%

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

8. Simplified 6.3%

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

9. Taylor expanded in cosTheta_O around inf

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

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

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

    2. *-lowering-*.f32 40.5%

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

11. Simplified 40.5%

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

Alternative 9: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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 13.5%

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

5. Simplified 13.5%

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

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 2024162 
(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))))))