HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 11.2s

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: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{cosTheta\_O \cdot \left(\frac{0.6931 - \frac{1}{v}}{cosTheta\_O} + \frac{cosTheta\_i}{v}\right)} \cdot 0.5}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (exp
    (* cosTheta_O (+ (/ (- 0.6931 (/ 1.0 v)) cosTheta_O) (/ cosTheta_i v))))
   0.5)
  v))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp((costheta_o * (((0.6931e0 - (1.0e0 / v)) / costheta_o) + (costheta_i / v)))) * 0.5e0) / v
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp((cosTheta_O * (((single(0.6931) - (single(1.0) / v)) / cosTheta_O) + (cosTheta_i / v)))) * single(0.5)) / 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-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in cosTheta_O around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f32 N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. associate--r+ N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-*.f32 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in sinTheta_i around 0

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

10. Applied rewrites 99.5%

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

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. associate-*l/ N/A

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

    4. lower-/.f32 N/A

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

12. Applied rewrites 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 18.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -3.3800000551087635 \cdot 10^{-36}:\\ \;\;\;\;e^{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}\\ \mathbf{else}:\\ \;\;\;\;e^{sinTheta\_i \cdot \left(sinTheta\_O \cdot \frac{-1}{v}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -3.3800000551087635e-36)
   (exp (* (/ cosTheta_O v) cosTheta_i))
   (exp (* sinTheta_i (* sinTheta_O (/ -1.0 v))))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -3.3800000551087635e-36f) {
		tmp = expf(((cosTheta_O / v) * cosTheta_i));
	} else {
		tmp = expf((sinTheta_i * (sinTheta_O * (-1.0f / v))));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((costheta_i * costheta_o) <= (-3.3800000551087635e-36)) then
        tmp = exp(((costheta_o / v) * costheta_i))
    else
        tmp = exp((sintheta_i * (sintheta_o * ((-1.0e0) / v))))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-3.3800000551087635e-36))
		tmp = exp(Float32(Float32(cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp(Float32(sinTheta_i * Float32(sinTheta_O * Float32(Float32(-1.0) / v))));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-3.3800000551087635e-36))
		tmp = exp(((cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp((sinTheta_i * (sinTheta_O * (single(-1.0) / v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -3.38000006e-36

   1. Initial program 99.8%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in cosTheta_i around inf

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

   8. Applied rewrites 37.7%

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

   if -3.38000006e-36 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.7%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in sinTheta_i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 14.0

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

   8. Applied rewrites 14.0%

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

   10. Applied rewrites 14.0%

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

   12. Applied rewrites 14.0%

       \[\leadsto e^{sinTheta\_i \cdot \left(sinTheta\_O \cdot \color{blue}{\frac{-1}{v}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{0.5}{v} \cdot e^{0.6931 + \frac{-1}{v}} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* (/ 0.5 v) (exp (+ 0.6931 (/ -1.0 v)))))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 / v) * exp((0.6931e0 + ((-1.0e0) / v)))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) / v) * exp((single(0.6931) + (single(-1.0) / v)));
end

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. exp-sum N/A

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

   5. lift-log.f32 N/A

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

   6. rem-exp-log N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-exp.f32 99.8

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

   14. lift-+.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f32 99.1

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

7. Applied rewrites 97.6%

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

8. Taylor expanded in cosTheta_i around 0

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

10. Applied rewrites 99.8%

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

Alternative 4: 18.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -3.3800000551087635 \cdot 10^{-36}:\\ \;\;\;\;e^{\frac{cosTheta\_O}{v} \cdot cosTheta\_i}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-sinTheta\_i\right) \cdot \frac{sinTheta\_O}{v}}\\ \end{array} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -3.3800000551087635e-36)
   (exp (* (/ cosTheta_O v) cosTheta_i))
   (exp (* (- sinTheta_i) (/ sinTheta_O v)))))

C

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -3.3800000551087635e-36f) {
		tmp = expf(((cosTheta_O / v) * cosTheta_i));
	} else {
		tmp = expf((-sinTheta_i * (sinTheta_O / v)));
	}
	return tmp;
}

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if ((costheta_i * costheta_o) <= (-3.3800000551087635e-36)) then
        tmp = exp(((costheta_o / v) * costheta_i))
    else
        tmp = exp((-sintheta_i * (sintheta_o / v)))
    end if
    code = tmp
end function

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-3.3800000551087635e-36))
		tmp = exp(Float32(Float32(cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp(Float32(Float32(-sinTheta_i) * Float32(sinTheta_O / v)));
	end
	return tmp
end

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-3.3800000551087635e-36))
		tmp = exp(((cosTheta_O / v) * cosTheta_i));
	else
		tmp = exp((-sinTheta_i * (sinTheta_O / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -3.38000006e-36

   1. Initial program 99.8%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in cosTheta_i around inf

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

   8. Applied rewrites 37.7%

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

   if -3.38000006e-36 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.7%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in sinTheta_i around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 14.0

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

   8. Applied rewrites 14.0%

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

   10. Applied rewrites 14.0%

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

Alternative 5: 49.5% accurate, 2.2× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (/ (- (fma cosTheta_i cosTheta_O -1.0) (* sinTheta_i sinTheta_O)) v)))

C

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

Julia

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(Float32(fma(cosTheta_i, cosTheta_O, Float32(-1.0)) - Float32(sinTheta_i * sinTheta_O)) / v))
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 94.7%

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

6. Taylor expanded in cosTheta_i around inf

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

8. Applied rewrites 13.9%

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

9. Taylor expanded in v around 0

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

    1. lower-/.f32 N/A

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

    2. associate--r+ N/A

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

    3. lower--.f32 N/A

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

    4. sub-neg N/A

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

    5. *-commutative N/A

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

    6. metadata-eval N/A

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

    7. lower-fma.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 97.4

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

11. Applied rewrites 97.4%

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

Alternative 6: 83.9% accurate, 2.2× speedup

Math

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

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(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

assert(cosTheta_i < cosTheta_O && cosTheta_O < sinTheta_i && sinTheta_i < sinTheta_O && sinTheta_O < v);
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

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])
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. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 94.7%

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

6. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. associate--r+ N/A

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. 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 97.4

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

8. Applied rewrites 97.8%

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

Alternative 7: 13.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ e^{\frac{cosTheta\_O}{v} \cdot cosTheta\_i} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp (* (/ cosTheta_O v) cosTheta_i)))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = exp(((costheta_o / v) * costheta_i))
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = exp(((cosTheta_O / v) * cosTheta_i));
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in cosTheta_i around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 94.7%

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

6. Taylor expanded in cosTheta_i around inf

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

8. Applied rewrites 13.9%

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

Alternative 8: 4.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} [cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v] = \mathsf{sort}([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])\\ \\ \frac{e^{0.6931} \cdot 0.5}{v} \end{array} \]

FPCore

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* (exp 0.6931) 0.5) v))

C

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

Fortran

NOTE: cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, and v should be sorted in increasing order before calling this function.
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) * 0.5e0) / v
end function

Julia

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

MATLAB

cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v = num2cell(sort([cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v])){:}
function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(single(0.6931)) * single(0.5)) / 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%

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

Reproduce

herbie shell --seed 2024322 
(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))))))