HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 13.8s

Alternatives: 10

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: 99.7% 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: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

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

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

   5. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

    1. sub-neg N/A

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

    2. +-commutative N/A

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

    3. +-commutative N/A

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

    4. distribute-neg-in N/A

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

    5. distribute-frac-neg2 N/A

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

    6. metadata-eval N/A

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

    7. frac-2neg N/A

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

    8. metadata-eval N/A

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

    9. div-inv N/A

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

    10. metadata-eval N/A

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

    11. *-commutative N/A

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

    12. log-prod N/A

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

    13. associate-+r+ N/A

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

    14. exp-sum N/A

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

    15. rem-exp-log N/A

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

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

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / expf((logf((v / 0.5f)) - (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 = 1.0e0 / exp((log((v / 0.5e0)) - (0.6931e0 + ((-1.0e0) / v))))
end function

Julia

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

Derivation

1. Initial program 99.8%

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

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

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

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

   5. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

Alternative 3: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf(((0.6931f + (-1.0f / v)) - logf((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(((0.6931e0 + ((-1.0e0) / v)) - log((v / 0.5e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. rec-exp N/A

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

   2. neg-mul-1 N/A

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

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

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

   4. neg-mul-1 N/A

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

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

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

   6. associate--l+ N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

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

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

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

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

   13. metadata-eval 99.8%

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

10. Simplified 99.8%

    \[\leadsto \color{blue}{e^{-\left(\log \left(2 \cdot v\right) + \left(\frac{1}{v} + -0.6931\right)\right)}} \]
11. Step-by-step derivation

    1. neg-sub0 N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(0 - \left(\log \left(2 \cdot v\right) + \left(\frac{1}{v} + \frac{-6931}{10000}\right)\right)\right)\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(0 - \left(\left(\frac{1}{v} + \frac{-6931}{10000}\right) + \log \left(2 \cdot v\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(0 - \left(\left(\frac{1}{v} + \frac{-6931}{10000}\right) + \log \left(v \cdot 2\right)\right)\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(0 - \left(\left(\frac{1}{v} + \frac{-6931}{10000}\right) + \log \left(v \cdot \frac{1}{\frac{1}{2}}\right)\right)\right)\right) \]

    5. div-inv N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(0 - \left(\left(\frac{1}{v} + \frac{-6931}{10000}\right) + \log \left(\frac{v}{\frac{1}{2}}\right)\right)\right)\right) \]

    6. associate--r+ N/A

       \[\leadsto \mathsf{exp.f32}\left(\left(\left(0 - \left(\frac{1}{v} + \frac{-6931}{10000}\right)\right) - \log \left(\frac{v}{\frac{1}{2}}\right)\right)\right) \]

    7. neg-sub0 N/A

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

    8. distribute-neg-in N/A

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

    9. mul-1-neg N/A

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

    10. div-inv N/A

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

    11. metadata-eval N/A

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

    12. +-commutative N/A

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

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

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

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

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

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

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

    16. rem-exp-log N/A

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

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

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

    18. rem-exp-log N/A

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

    19. /-lowering-/.f32 99.8%

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

12. Applied egg-rr 99.8%

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

Alternative 4: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

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

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

   5. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

    1. /-rgt-identity N/A

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

    2. exp-diff N/A

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

    3. associate-/l/ N/A

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

    4. rem-exp-log N/A

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

    5. div-inv N/A

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

    6. metadata-eval N/A

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

    7. times-frac N/A

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

    8. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

    13. /-lowering-/.f32 99.8%

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

12. Applied egg-rr 99.8%

    \[\leadsto \frac{1}{\color{blue}{v \cdot \frac{2}{e^{0.6931 + \frac{-1}{v}}}}} \]
13. Add Preprocessing

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

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

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

   8. sub-neg N/A

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

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

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

   10. distribute-neg-in N/A

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

   11. sub-neg N/A

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

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

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. *-lowering-*.f32 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. /-lowering-/.f32 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 6: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

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

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

   8. sub-neg N/A

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

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

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

   10. distribute-neg-in N/A

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

   11. sub-neg N/A

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

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

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. *-lowering-*.f32 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. /-lowering-/.f32 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in v around 0

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

    1. /-lowering-/.f32 98.3%

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

11. Simplified 98.3%

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

Alternative 7: 98.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. metadata-eval N/A

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

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

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

   5. /-lowering-/.f32 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in v around 0

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

    1. /-lowering-/.f32 98.1%

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

13. Simplified 98.1%

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

Alternative 8: 98.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. associate-+l- N/A

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

   4. exp-diff N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{e^{0}}{e^{\color{blue}{\log \left(2 \cdot v\right)} - \left(\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}\right)}} \]

   6. 1-exp N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. *-lowering-*.f32 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in cosTheta_O around 0

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

   1. rec-exp N/A

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

   2. neg-mul-1 N/A

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

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

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

   4. neg-mul-1 N/A

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

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

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

   6. associate--l+ N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

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

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

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

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

   13. metadata-eval 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in v around 0

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

    1. /-lowering-/.f32 98.1%

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

13. Simplified 98.1%

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

Alternative 9: 38.1% 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 99.8%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

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

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

   4. /-lowering-/.f32 11.7%

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

5. Simplified 11.7%

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

6. Taylor expanded in cosTheta_i 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 6.2%

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

8. Simplified 6.2%

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

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 34.3%

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

11. Simplified 34.3%

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

Alternative 10: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

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

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

   4. /-lowering-/.f32 11.7%

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

5. Simplified 11.7%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Simplified 6.4%

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

Reproduce

herbie shell --seed 2024191 
(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))))))