HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 17.5s

Alternatives: 11

Speedup: 1.1×

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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+l+ N/A

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

   4. lift--.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. sub-div N/A

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

   9. lift-/.f32 N/A

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

   10. frac-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. lift-/.f32 N/A

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

   15. lift-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift--.f32 N/A

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

   4. associate-+r- N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. log-prod N/A

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

   8. associate--r+ N/A

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

   9. exp-diff N/A

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

9. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{e^{\left(0.6931 + \frac{\mathsf{fma}\left(cosTheta\_i \cdot cosTheta\_O, v, -v\right)}{v \cdot v}\right) - \log v}}{e^{\log 2}}} \]
10. Step-by-step derivation

    1. lift--.f32 N/A

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

    2. lift-+.f32 N/A

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

    3. +-commutative N/A

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

    4. associate--l+ N/A

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

    5. lift-/.f32 N/A

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

    6. div-inv N/A

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

    7. lift-/.f32 N/A

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

    8. lower-fma.f32 N/A

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

    9. lower--.f32 99.8

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate-+l+ N/A

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

   4. lift--.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. sub-div N/A

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

   9. lift-/.f32 N/A

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

   10. frac-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. frac-times N/A

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

   14. lift-/.f32 N/A

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

   15. lift-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in cosTheta_i around inf

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

   1. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift--.f32 N/A

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

   4. associate-+r- N/A

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

   5. lift-log.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. log-prod N/A

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

   8. associate--r+ N/A

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

   9. exp-diff N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in sinTheta_i around 0

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

    1. lower-*.f32 N/A

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

    2. lower-exp.f32 N/A

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

    3. lower--.f32 N/A

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

    4. +-commutative N/A

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

    5. associate-/l* N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. lower-+.f32 N/A

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

    9. lower-log.f32 N/A

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

    10. lower-/.f32 99.8

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

12. Applied rewrites 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 22.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{elif}\;cosTheta\_i \cdot cosTheta\_O \leq 3.0000000565330046 \cdot 10^{-31}:\\ \;\;\;\;e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{-t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_i cosTheta_O) v)))
   (if (<= (* cosTheta_i cosTheta_O) -4.000000072010038e-35)
     (exp t_0)
     (if (<= (* cosTheta_i cosTheta_O) 3.0000000565330046e-31)
       (exp (- (/ (* sinTheta_i sinTheta_O) v)))
       (exp (- t_0))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) / v;
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -4.000000072010038e-35f) {
		tmp = expf(t_0);
	} else if ((cosTheta_i * cosTheta_O) <= 3.0000000565330046e-31f) {
		tmp = expf(-((sinTheta_i * sinTheta_O) / v));
	} 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_i * costheta_o) / v
    if ((costheta_i * costheta_o) <= (-4.000000072010038e-35)) then
        tmp = exp(t_0)
    else if ((costheta_i * costheta_o) <= 3.0000000565330046e-31) then
        tmp = exp(-((sintheta_i * sintheta_o) / v))
    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_i * cosTheta_O) / v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-4.000000072010038e-35))
		tmp = exp(t_0);
	elseif (Float32(cosTheta_i * cosTheta_O) <= Float32(3.0000000565330046e-31))
		tmp = exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v)));
	else
		tmp = exp(Float32(-t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_i * cosTheta_O) / v;
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-4.000000072010038e-35))
		tmp = exp(t_0);
	elseif ((cosTheta_i * cosTheta_O) <= single(3.0000000565330046e-31))
		tmp = exp(-((sinTheta_i * sinTheta_O) / v));
	else
		tmp = exp(-t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -4.00000007e-35

   1. Initial program 99.9%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 44.1

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

   5. Applied rewrites 44.1%

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

   7. Applied rewrites 44.1%

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

   if -4.00000007e-35 < (*.f32 cosTheta_i cosTheta_O) < 3.00000006e-31

   1. Initial program 99.7%

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

   3. Taylor expanded in sinTheta_i around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 18.3

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

   5. Applied rewrites 18.3%

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

   if 3.00000006e-31 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.7%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 4.5

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

   5. Applied rewrites 4.5%

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

   7. Applied rewrites 60.0%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{elif}\;cosTheta\_i \cdot cosTheta\_O \leq 3.0000000565330046 \cdot 10^{-31}:\\ \;\;\;\;e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 22.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_i\right) \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -2.0000000063421537e-30)
   (exp (* sinTheta_i (/ sinTheta_O v)))
   (if (<= (* sinTheta_i sinTheta_O) 4.999999898305949e-32)
     (exp (/ (* cosTheta_i cosTheta_O) v))
     (exp (* (- sinTheta_i) (/ sinTheta_O v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -2.0000000063421537e-30f) {
		tmp = expf((sinTheta_i * (sinTheta_O / v)));
	} else if ((sinTheta_i * sinTheta_O) <= 4.999999898305949e-32f) {
		tmp = expf(((cosTheta_i * cosTheta_O) / v));
	} else {
		tmp = expf((-sinTheta_i * (sinTheta_O / 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 ((sintheta_i * sintheta_o) <= (-2.0000000063421537e-30)) then
        tmp = exp((sintheta_i * (sintheta_o / v)))
    else if ((sintheta_i * sintheta_o) <= 4.999999898305949e-32) then
        tmp = exp(((costheta_i * costheta_o) / v))
    else
        tmp = exp((-sintheta_i * (sintheta_o / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(-2.0000000063421537e-30))
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / v)));
	elseif (Float32(sinTheta_i * sinTheta_O) <= Float32(4.999999898305949e-32))
		tmp = exp(Float32(Float32(cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(Float32(Float32(-sinTheta_i) * Float32(sinTheta_O / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(-2.0000000063421537e-30))
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	elseif ((sinTheta_i * sinTheta_O) <= single(4.999999898305949e-32))
		tmp = exp(((cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp((-sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -2e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 18.5

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in sinTheta_i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 4.3

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 65.6%

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

   if -2e-30 < (*.f32 sinTheta_i sinTheta_O) < 4.9999999e-32

   1. Initial program 99.6%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 14.2

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

   5. Applied rewrites 14.2%

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

   7. Applied rewrites 14.2%

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

   if 4.9999999e-32 < (*.f32 sinTheta_i sinTheta_O)

   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

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 14.0

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in sinTheta_i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 57.4

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

   8. Applied rewrites 57.4%

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

   10. Applied rewrites 57.4%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_i\right) \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 22.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -2.0000000063421537e-30)
   (exp (* sinTheta_i (/ sinTheta_O v)))
   (if (<= (* sinTheta_i sinTheta_O) 4.999999898305949e-32)
     (exp (/ (* cosTheta_i cosTheta_O) v))
     (exp (- (/ (* sinTheta_i sinTheta_O) v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -2.0000000063421537e-30f) {
		tmp = expf((sinTheta_i * (sinTheta_O / v)));
	} else if ((sinTheta_i * sinTheta_O) <= 4.999999898305949e-32f) {
		tmp = expf(((cosTheta_i * cosTheta_O) / v));
	} else {
		tmp = expf(-((sinTheta_i * sinTheta_O) / 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 ((sintheta_i * sintheta_o) <= (-2.0000000063421537e-30)) then
        tmp = exp((sintheta_i * (sintheta_o / v)))
    else if ((sintheta_i * sintheta_o) <= 4.999999898305949e-32) then
        tmp = exp(((costheta_i * costheta_o) / v))
    else
        tmp = exp(-((sintheta_i * sintheta_o) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(-2.0000000063421537e-30))
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / v)));
	elseif (Float32(sinTheta_i * sinTheta_O) <= Float32(4.999999898305949e-32))
		tmp = exp(Float32(Float32(cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(-2.0000000063421537e-30))
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	elseif ((sinTheta_i * sinTheta_O) <= single(4.999999898305949e-32))
		tmp = exp(((cosTheta_i * cosTheta_O) / v));
	else
		tmp = exp(-((sinTheta_i * sinTheta_O) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -2e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 18.5

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in sinTheta_i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 4.3

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 65.6%

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

   if -2e-30 < (*.f32 sinTheta_i sinTheta_O) < 4.9999999e-32

   1. Initial program 99.6%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 14.2

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

   5. Applied rewrites 14.2%

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

   7. Applied rewrites 14.2%

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

   if 4.9999999e-32 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.8%

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

   3. Taylor expanded in sinTheta_i around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 57.4

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

   5. Applied rewrites 57.4%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{elif}\;sinTheta\_i \cdot sinTheta\_O \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_i around 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

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. sub-neg N/A

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

   9. lower-+.f32 N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

   19. lower-*.f32 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in sinTheta_O around 0

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

8. Applied rewrites 99.7%

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

Alternative 7: 17.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -2.0000000063421537e-30)
   (exp (* sinTheta_i (/ sinTheta_O v)))
   (exp (/ (* cosTheta_i cosTheta_O) v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_i * sinTheta_O) <= -2.0000000063421537e-30f) {
		tmp = expf((sinTheta_i * (sinTheta_O / v)));
	} else {
		tmp = expf(((cosTheta_i * cosTheta_O) / 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 ((sintheta_i * sintheta_o) <= (-2.0000000063421537e-30)) then
        tmp = exp((sintheta_i * (sintheta_o / v)))
    else
        tmp = exp(((costheta_i * costheta_o) / v))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(sinTheta_i * sinTheta_O) <= Float32(-2.0000000063421537e-30))
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / v)));
	else
		tmp = exp(Float32(Float32(cosTheta_i * cosTheta_O) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_i * sinTheta_O) <= single(-2.0000000063421537e-30))
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	else
		tmp = exp(((cosTheta_i * cosTheta_O) / v));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -2e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 18.5

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in sinTheta_i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 4.3

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 65.6%

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

   if -2e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 14.1

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

   5. Applied rewrites 14.1%

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

   7. Applied rewrites 14.1%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 17.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_i sinTheta_O) -2.0000000063421537e-30)
   (exp (* sinTheta_i (/ sinTheta_O 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 ((sinTheta_i * sinTheta_O) <= -2.0000000063421537e-30f) {
		tmp = expf((sinTheta_i * (sinTheta_O / 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 ((sintheta_i * sintheta_o) <= (-2.0000000063421537e-30)) then
        tmp = exp((sintheta_i * (sintheta_o / 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 (Float32(sinTheta_i * sinTheta_O) <= Float32(-2.0000000063421537e-30))
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O / 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 ((sinTheta_i * sinTheta_O) <= single(-2.0000000063421537e-30))
		tmp = exp((sinTheta_i * (sinTheta_O / v)));
	else
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < -2e-30

   1. Initial program 100.0%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 18.5

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

   5. Applied rewrites 18.5%

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

   6. Taylor expanded in sinTheta_i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 4.3

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

   8. Applied rewrites 4.3%

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

   10. Applied rewrites 65.6%

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

   if -2e-30 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.7%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f32 N/A

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

      4. lower-/.f32 14.1

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

   5. Applied rewrites 14.1%

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

   6. Taylor expanded in cosTheta_i around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-/.f32 14.1

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

   8. Applied rewrites 14.1%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_i \cdot sinTheta\_O \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;e^{sinTheta\_i \cdot \frac{sinTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. distribute-neg-in N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. mul-1-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f32 97.8

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

5. Applied rewrites 97.8%

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

Alternative 10: 12.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((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((costheta_o * (costheta_i / v)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 14.6

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

5. Applied rewrites 14.6%

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

6. Taylor expanded in cosTheta_i around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 14.6

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

8. Applied rewrites 14.6%

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

Alternative 11: 6.4% accurate, 272.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in cosTheta_i around inf

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

7. Applied rewrites 6.5%

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

Reproduce

herbie shell --seed 2024227 
(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))))))