HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 11.8s

Alternatives: 8

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.5

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

5. Applied rewrites 40.8%

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

7. Applied rewrites 43.4%

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

8. Taylor expanded in cosTheta_i around 0

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

10. Applied rewrites 99.7%

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

12. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 41.2

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

5. Applied rewrites 40.8%

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

7. Applied rewrites 42.3%

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

8. Taylor expanded in cosTheta_i around 0

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

10. Applied rewrites 99.7%

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

12. Applied rewrites 99.8%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.8

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

5. Applied rewrites 41.2%

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

7. Applied rewrites 43.0%

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

8. Taylor expanded in cosTheta_i around 0

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

10. Applied rewrites 99.7%

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

Alternative 4: 99.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 40.8

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

5. Applied rewrites 40.8%

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in cosTheta_i around 0

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

10. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. associate-+r+ N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. exp-sum N/A

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

   7. lower-*.f32 N/A

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

   8. rem-exp-log N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 41.2

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

5. Applied rewrites 41.2%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Applied rewrites 99.7%

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

Alternative 6: 97.7% 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 99.7%

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

   1. lift-+.f32 N/A

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

   2. flip3-+ N/A

      \[\leadsto e^{\color{blue}{\frac{{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)}^{3} + {\frac{6931}{10000}}^{3}}{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) \cdot \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(\frac{6931}{10000} \cdot \frac{6931}{10000} - \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) \cdot \frac{6931}{10000}\right)}} + \log \left(\frac{1}{2 \cdot v}\right)} \]

   3. clear-num N/A

      \[\leadsto e^{\color{blue}{\frac{1}{\frac{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) \cdot \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(\frac{6931}{10000} \cdot \frac{6931}{10000} - \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) \cdot \frac{6931}{10000}\right)}{{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)}^{3} + {\frac{6931}{10000}}^{3}}}} + \log \left(\frac{1}{2 \cdot v}\right)} \]

   4. lower-/.f32 N/A

      \[\leadsto e^{\color{blue}{\frac{1}{\frac{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) \cdot \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \left(\frac{6931}{10000} \cdot \frac{6931}{10000} - \left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) \cdot \frac{6931}{10000}\right)}{{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right)}^{3} + {\frac{6931}{10000}}^{3}}}} + \log \left(\frac{1}{2 \cdot v}\right)} \]

4. Applied rewrites 99.7%

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

5. Taylor expanded in sinTheta_O around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-+.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. log-rec N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in v around 0

    \[\leadsto e^{\color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i - \left(1 + sinTheta\_O \cdot sinTheta\_i\right)}{v}}} \]
11. 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 96.6

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

12. Applied rewrites 95.9%

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

Alternative 7: 4.6% accurate, 2.3× speedup

Math

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

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(Float32(0.5) * 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 99.7%

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

3. Taylor expanded in v around -inf

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

   1. associate-+r+ N/A

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

   2. exp-sum N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. rem-exp-log N/A

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

   6. lower-*.f32 N/A

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

   7. exp-sum N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 4.7

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

5. Applied rewrites 4.7%

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

7. Applied rewrites 4.7%

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

Alternative 8: 4.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around -inf

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

   1. associate-+r+ N/A

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

   2. exp-sum N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. rem-exp-log N/A

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

   6. lower-*.f32 N/A

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

   7. exp-sum N/A

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

   8. rem-exp-log N/A

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

   9. lower-*.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 4.7

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

5. Applied rewrites 4.7%

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

7. Applied rewrites 4.7%

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

Reproduce

herbie shell --seed 2024324 
(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))))))