HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 12.7s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f32 N/A

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

   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. lift-+.f32 N/A

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. associate--l- N/A

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

   12. lower--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 40.0

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

6. Applied rewrites 40.0%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. mul-1-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. lower-/.f32 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in cosTheta_O around 0

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

12. Applied rewrites 99.8%

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

13. Final simplification 99.8%

    \[\leadsto \frac{1}{e^{\frac{1}{v}}} \cdot \left(\frac{0.5}{v} \cdot e^{0.6931}\right) \]
14. Add Preprocessing

Alternative 2: 18.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_i \leq 1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;e^{\left(\frac{1}{v} \cdot cosTheta\_O\right) \cdot cosTheta\_i}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_i) 1.999999982195158e-37)
   (exp (* (* (/ 1.0 v) cosTheta_O) cosTheta_i))
   (exp (* (/ (- sinTheta_i) v) sinTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_O * sinTheta_i) <= 1.999999982195158e-37f) {
		tmp = expf((((1.0f / v) * cosTheta_O) * cosTheta_i));
	} else {
		tmp = expf(((-sinTheta_i / v) * sinTheta_O));
	}
	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_o * sintheta_i) <= 1.999999982195158e-37) then
        tmp = exp((((1.0e0 / v) * costheta_o) * costheta_i))
    else
        tmp = exp(((-sintheta_i / v) * sintheta_o))
    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_O * sinTheta_i) <= Float32(1.999999982195158e-37))
		tmp = exp(Float32(Float32(Float32(Float32(1.0) / v) * cosTheta_O) * cosTheta_i));
	else
		tmp = exp(Float32(Float32(Float32(-sinTheta_i) / v) * sinTheta_O));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_O * sinTheta_i) <= single(1.999999982195158e-37))
		tmp = exp((((single(1.0) / v) * cosTheta_O) * cosTheta_i));
	else
		tmp = exp(((-sinTheta_i / v) * sinTheta_O));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 1.99999998e-37

   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_O around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 13.1

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

   5. Applied rewrites 13.1%

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

   7. Applied rewrites 13.1%

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

   9. Applied rewrites 13.1%

      \[\leadsto e^{\left(\frac{1}{v} \cdot cosTheta\_O\right) \cdot cosTheta\_i} \]

   if 1.99999998e-37 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.4%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-/.f32 33.8

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

   5. Applied rewrites 33.8%

      \[\leadsto e^{\color{blue}{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_i \leq 1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;e^{\left(\frac{1}{v} \cdot cosTheta\_O\right) \cdot cosTheta\_i}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.1× 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. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   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. +-commutative N/A

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in cosTheta_O around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-neg-in N/A

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

   11. +-commutative N/A

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

   12. mul-1-neg N/A

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

   13. associate-*r/ N/A

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

   14. mul-1-neg N/A

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

   15. unsub-neg N/A

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

   16. lower--.f32 N/A

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

   17. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 18.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_i \leq 1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_i) 1.999999982195158e-37)
   (exp (/ (* cosTheta_O cosTheta_i) v))
   (exp (* (/ (- sinTheta_i) v) sinTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((sinTheta_O * sinTheta_i) <= 1.999999982195158e-37f) {
		tmp = expf(((cosTheta_O * cosTheta_i) / v));
	} else {
		tmp = expf(((-sinTheta_i / v) * sinTheta_O));
	}
	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_o * sintheta_i) <= 1.999999982195158e-37) then
        tmp = exp(((costheta_o * costheta_i) / v))
    else
        tmp = exp(((-sintheta_i / v) * sintheta_o))
    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_O * sinTheta_i) <= Float32(1.999999982195158e-37))
		tmp = exp(Float32(Float32(cosTheta_O * cosTheta_i) / v));
	else
		tmp = exp(Float32(Float32(Float32(-sinTheta_i) / v) * sinTheta_O));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((sinTheta_O * sinTheta_i) <= single(1.999999982195158e-37))
		tmp = exp(((cosTheta_O * cosTheta_i) / v));
	else
		tmp = exp(((-sinTheta_i / v) * sinTheta_O));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_i sinTheta_O) < 1.99999998e-37

   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_O around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 13.1

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

   5. Applied rewrites 13.1%

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

   if 1.99999998e-37 < (*.f32 sinTheta_i sinTheta_O)

   1. Initial program 99.4%

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

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-/.f32 33.8

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

   5. Applied rewrites 33.8%

      \[\leadsto e^{\color{blue}{\left(-sinTheta\_O\right) \cdot \frac{sinTheta\_i}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_i \leq 1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;e^{\frac{cosTheta\_O \cdot cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-sinTheta\_i}{v} \cdot sinTheta\_O}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) - Float32(sinTheta_O * sinTheta_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 sinTheta_i around inf

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-/.f32 11.7

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

5. Applied rewrites 11.7%

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

6. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
7. 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. associate--r+ N/A

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.5

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

8. Applied rewrites 97.5%

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

9. Final simplification 97.8%

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

Alternative 6: 84.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(fma(cosTheta_O, cosTheta_i, Float32(-1.0)) - Float32(sinTheta_O * sinTheta_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 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. associate--r+ N/A

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 97.5

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

5. Applied rewrites 97.5%

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

Alternative 7: 13.3% accurate, 2.3× speedup

Math

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

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(Float32(cosTheta_O * 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 99.8%

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

3. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 12.5

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

5. Applied rewrites 12.5%

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

Alternative 8: 13.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 12.5

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

5. Applied rewrites 12.5%

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

7. Applied rewrites 12.5%

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

Alternative 9: 13.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 / v) * costheta_i))
end function

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 12.5

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

5. Applied rewrites 12.5%

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

7. Applied rewrites 12.5%

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

Alternative 10: 4.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (exp(single(-0.6931)) * 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. lift-exp.f32 N/A

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

   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. +-commutative N/A

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 4.7

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

7. Applied rewrites 4.7%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v} \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 4.7%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{v}{e^{0.6931}}}} \]
10. Step-by-step derivation

11. Applied rewrites 4.7%

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

Alternative 11: 4.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(0.6931f) / 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) / v) * 0.5e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(single(0.6931)) / 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. lift-exp.f32 N/A

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

   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. +-commutative N/A

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 4.7

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

7. Applied rewrites 4.7%

   \[\leadsto \color{blue}{\frac{e^{0.6931}}{v} \cdot 0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))))