HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 9.5s

Alternatives: 7

Speedup: 1.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. log-rec N/A

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

   5. lift-log.f32 N/A

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

   6. unsub-neg N/A

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

   7. lower--.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift--.f32 N/A

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

   3. exp-diff N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 99.7

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in cosTheta_i around inf

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

    1. lower-/.f32 N/A

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

    2. lower-*.f32 99.7

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

12. Applied rewrites 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((-sinTheta_O * sinTheta_i) / v) - ((1.0f / v) - 0.6931f))) * (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((((-sintheta_o * sintheta_i) / v) - ((1.0e0 / v) - 0.6931e0))) * (0.5e0 / v)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. associate-+l- N/A

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

   7. associate-+r- N/A

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

7. Applied rewrites 90.0%

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

8. Taylor expanded in v around -inf

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. rem-exp-log N/A

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

   4. lower-*.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 99.3

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

10. Applied rewrites 99.3%

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

Alternative 4: 97.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. log-rec N/A

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

   5. lift-log.f32 N/A

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

   6. unsub-neg N/A

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

   7. lower--.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
9. 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 97.1

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

10. Applied rewrites 97.1%

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

11. Taylor expanded in cosTheta_i around 0

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

13. Applied rewrites 97.5%

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

Alternative 5: 97.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ (fma cosTheta_O cosTheta_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, -1.0f) / v));
}

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in cosTheta_i around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-/.f32 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. log-rec N/A

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

   5. lift-log.f32 N/A

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

   6. unsub-neg N/A

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

   7. lower--.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in v around 0

   \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
9. 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 97.1

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

10. Applied rewrites 97.1%

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

11. Taylor expanded in sinTheta_i around 0

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

13. Applied rewrites 97.1%

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

Alternative 6: 10.7% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (* (exp 0.6931) v) 2.0))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. flip-+ N/A

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

   5. clear-num-rev N/A

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

   6. +-inverses N/A

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

   7. +-inverses N/A

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

   8. +-inverses N/A

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

   9. +-inverses N/A

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

   10. flip-+ N/A

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

   11. count-2-rev N/A

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

   12. lift-*.f32 97.5

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. exp-sum N/A

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

   4. rem-exp-log N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. remove-double-neg N/A

      \[\leadsto \left(2 \cdot e^{\color{blue}{\log v}}\right) \cdot e^{\frac{6931}{10000}} \]

   8. rem-exp-log N/A

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

   9. rem-exp-log N/A

      \[\leadsto \color{blue}{e^{\log \left(2 \cdot v\right)}} \cdot e^{\frac{6931}{10000}} \]

   10. lower-*.f32 N/A

       \[\leadsto \color{blue}{e^{\log \left(2 \cdot v\right)} \cdot e^{\frac{6931}{10000}}} \]

   11. rem-exp-log N/A

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

   12. lower-*.f32 N/A

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

   13. lower-exp.f32 10.5

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

7. Applied rewrites 10.5%

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

9. Applied rewrites 10.5%

   \[\leadsto \color{blue}{\left(e^{0.6931} \cdot v\right) \cdot 2} \]
10. Add Preprocessing

Alternative 7: 10.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. flip-+ N/A

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

   5. clear-num-rev N/A

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

   6. +-inverses N/A

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

   7. +-inverses N/A

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

   8. +-inverses N/A

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

   9. +-inverses N/A

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

   10. flip-+ N/A

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

   11. count-2-rev N/A

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

   12. lift-*.f32 97.5

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

4. Applied rewrites 97.5%

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

5. Taylor expanded in v around inf

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. exp-sum N/A

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

   4. rem-exp-log N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. remove-double-neg N/A

      \[\leadsto \left(2 \cdot e^{\color{blue}{\log v}}\right) \cdot e^{\frac{6931}{10000}} \]

   8. rem-exp-log N/A

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

   9. rem-exp-log N/A

      \[\leadsto \color{blue}{e^{\log \left(2 \cdot v\right)}} \cdot e^{\frac{6931}{10000}} \]

   10. lower-*.f32 N/A

       \[\leadsto \color{blue}{e^{\log \left(2 \cdot v\right)} \cdot e^{\frac{6931}{10000}}} \]

   11. rem-exp-log N/A

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

   12. lower-*.f32 N/A

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

   13. lower-exp.f32 10.5

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

7. Applied rewrites 10.5%

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

9. Applied rewrites 10.5%

   \[\leadsto \left(v + v\right) \cdot e^{\color{blue}{0.6931}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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))))))