HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 16.6s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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

5. Taylor expanded in v around 0

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 99.7%

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

7. Simplified 99.7%

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

   1. frac-2neg N/A

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

   2. div-inv N/A

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

   3. exp-prod N/A

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

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

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

9. Applied egg-rr 99.8%

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

10. Taylor expanded in sinTheta_i around 0

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

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

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

    2. +-commutative N/A

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

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

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

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

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

    5. *-commutative N/A

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

    6. *-lowering-*.f32 99.8%

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

12. Simplified 99.8%

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

13. Taylor expanded in cosTheta_O around 0

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

    1. cancel-sign-sub-inv N/A

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

    2. metadata-eval N/A

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

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

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

    4. metadata-eval N/A

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

    5. cancel-sign-sub-inv N/A

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

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

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

    7. *-commutative N/A

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

    8. *-lowering-*.f32 99.8%

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

15. Simplified 99.8%

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

16. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (pow (exp (/ 1.0 v)) (+ (* v 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 * powf(expf((1.0f / v)), ((v * 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 * (exp((1.0e0 / v)) ** ((v * 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) * (exp(Float32(Float32(1.0) / v)) ^ Float32(Float32(v * Float32(0.6931)) + Float32(-1.0)))) / v)
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

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

5. Taylor expanded in v around 0

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. associate--r+ N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. *-lowering-*.f32 99.7%

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

10. Simplified 99.7%

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

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

    3. exp-prod N/A

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

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

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

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

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

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

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

    7. sub-neg N/A

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

    8. *-commutative N/A

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

    9. *-commutative N/A

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

    10. associate-+l+ N/A

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

    11. metadata-eval N/A

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

    12. distribute-neg-in N/A

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

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

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

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

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

    15. distribute-neg-in N/A

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

    16. metadata-eval N/A

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

    17. sub-neg N/A

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

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

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

    19. *-lowering-*.f32 99.7%

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

12. Applied egg-rr 99.7%

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

13. Taylor expanded in sinTheta_i around 0

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

    1. sub-neg N/A

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

    2. metadata-eval N/A

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

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

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

    4. *-commutative N/A

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

    5. *-lowering-*.f32 99.7%

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

15. Simplified 99.7%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ (+ -1.0 (* cosTheta_O cosTheta_i)) 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 + (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 = (0.5e0 / v) * exp((0.6931e0 + (((-1.0e0) + (costheta_o * costheta_i)) / 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(Float32(-1.0) + Float32(cosTheta_O * cosTheta_i)) / 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) + (cosTheta_O * cosTheta_i)) / v)));
end

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. rem-exp-log N/A

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

   4. +-commutative N/A

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

   5. associate--l- N/A

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

   6. associate-+r- N/A

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

   7. +-commutative N/A

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

   8. associate-+r- N/A

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

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

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

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. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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

5. Taylor expanded in v around 0

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

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

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. associate--r+ N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. *-lowering-*.f32 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in sinTheta_O around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{\frac{\frac{6931}{10000} \cdot v - 1}{v}}}{v}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

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

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

    9. *-commutative N/A

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

    10. *-lowering-*.f32 99.7%

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

13. Simplified 99.7%

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

Alternative 5: 40.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{if}\;cosTheta\_O \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_O cosTheta_i) v)))
   (if (<= cosTheta_O 1.9999999920083944e-11) t_0 (exp t_0))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_O * cosTheta_i) / v;
	float tmp;
	if (cosTheta_O <= 1.9999999920083944e-11f) {
		tmp = t_0;
	} else {
		tmp = expf(t_0);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (costheta_o * costheta_i) / v
    if (costheta_o <= 1.9999999920083944e-11) then
        tmp = t_0
    else
        tmp = exp(t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_O * cosTheta_i) / v)
	tmp = Float32(0.0)
	if (cosTheta_O <= Float32(1.9999999920083944e-11))
		tmp = t_0;
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_O * cosTheta_i) / v;
	tmp = single(0.0);
	if (cosTheta_O <= single(1.9999999920083944e-11))
		tmp = t_0;
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_O < 1.99999999e-11

   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. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 12.5%

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

   5. Simplified 12.5%

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

   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. +-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f32 6.3%

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

   8. Simplified 6.3%

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

   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 47.6%

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

   11. Simplified 47.6%

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

   if 1.99999999e-11 < 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. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 19.1%

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

   5. Simplified 19.1%

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

Alternative 6: 40.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_O 1.9999999920083944e-11)
   (/ (* 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 <= 1.9999999920083944e-11f) {
		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 <= 1.9999999920083944e-11) 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(1.9999999920083944e-11))
		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(1.9999999920083944e-11))
		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 < 1.99999999e-11

   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. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 12.5%

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

   5. Simplified 12.5%

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

   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. +-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f32 6.3%

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

   8. Simplified 6.3%

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

   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 47.6%

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

   11. Simplified 47.6%

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

   if 1.99999999e-11 < 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. /-lowering-/.f32 N/A

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

      2. *-lowering-*.f32 19.1%

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

   5. Simplified 19.1%

      \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}} \]
   6. 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. *-commutative N/A

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

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

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

      4. /-lowering-/.f32 19.1%

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

   7. Applied egg-rr 19.1%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_O \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in 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 sinTheta_O around 0

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

8. Simplified 97.8%

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

9. Final simplification 97.8%

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

Alternative 8: 38.7% 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 99.6%

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

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

   2. *-lowering-*.f32 13.6%

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

5. Simplified 13.6%

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

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. +-commutative N/A

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

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

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

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

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

   4. *-lowering-*.f32 6.3%

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

8. Simplified 6.3%

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

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 41.4%

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

11. Simplified 41.4%

    \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
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 99.6%

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

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

   2. *-lowering-*.f32 13.6%

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

5. Simplified 13.6%

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

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 2024288 
(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))))))