HairBSDF, Mp, lower

Percentage Accurate: 99.7% → 99.6%

Time: 16.7s

Alternatives: 10

Speedup: 2.0×

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: 99.7% 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: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(v / Float32(Float32(1.0) / (Float32(exp(1)) ^ Float32(Float32(-0.6931) + Float32(Float32(Float32(1.0) - Float32(cosTheta_O * cosTheta_i)) / v))))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (single(1.0) / (single(2.71828182845904523536) ^ (single(-0.6931) + ((single(1.0) - (cosTheta_O * cosTheta_i)) / v)))));
end

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.9%

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

   1. /-rgt-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \left(\frac{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}}}{\color{blue}{1}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \left(\frac{1}{\color{blue}{\frac{1}{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}}}}}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{1}{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}}}\right)}\right)\right)\right) \]

   4. rec-exp N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \left(e^{\mathsf{neg}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}\right)\right)}\right)\right)\right)\right) \]

   5. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{\mathsf{neg}\left(\left(cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)\right)\right)}{v} + \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{\mathsf{neg}\left(\left(cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)\right)\right)}{v}\right), \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{1}{v} - \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{1}{v} - \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right) \]

   2. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{v}\right), \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right)\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 99.9%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.9%

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

11. Applied egg-rr 99.9%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. log-rec N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right)\right), v\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right), v\right) \]

   15. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right), v\right) \]

7. Simplified 99.9%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{v}{e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)}}\right)}\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{\left(e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)}\right)}\right)\right) \]

   6. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \frac{-1}{v}\right) + cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\frac{-1}{v} + cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v} + cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \frac{-1}{v}\right)\right)\right)\right)\right) \]

   11. div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + -1 \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right)\right) \]

   13. unsub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(cosTheta\_O \cdot \frac{cosTheta\_i}{v} - \frac{1}{v}\right)\right)\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} - \frac{1}{v}\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{1}{v}\right)\right)\right)\right)\right) \]

   16. sub-div N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{cosTheta\_i \cdot cosTheta\_O - 1}{v}\right)\right)\right)\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O - 1\right), v\right)\right)\right)\right)\right) \]

   18. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O + \left(\mathsf{neg}\left(1\right)\right)\right), v\right)\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O + -1\right), v\right)\right)\right)\right)\right) \]

   20. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(cosTheta\_i \cdot cosTheta\_O\right), -1\right), v\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_i, cosTheta\_O\right), -1\right), v\right)\right)\right)\right)\right) \]

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (/ v (/ 1.0 (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 / (1.0f / 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 / (1.0e0 / exp(((1.0e0 / v) - 0.6931e0))))
end function

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.9%

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

   1. /-rgt-identity N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \left(\frac{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}}}{\color{blue}{1}}\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \left(\frac{1}{\color{blue}{\frac{1}{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}}}}}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{1}{e^{\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}}}\right)}\right)\right)\right) \]

   4. rec-exp N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \left(e^{\mathsf{neg}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}\right)\right)}\right)\right)\right)\right) \]

   5. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v} + \frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)}{v}\right)\right) + \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{\mathsf{neg}\left(\left(cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)\right)\right)}{v} + \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{\mathsf{neg}\left(\left(cosTheta\_i \cdot cosTheta\_O - \left(sinTheta\_i \cdot sinTheta\_O + 1\right)\right)\right)}{v}\right), \left(\mathsf{neg}\left(\frac{6931}{10000}\right)\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Taylor expanded in sinTheta_i around 0

   \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \color{blue}{\left(e^{\frac{1}{v} - \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\left(\frac{1}{v} - \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right) \]

   2. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{v}\right), \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)\right)\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right)\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 99.9%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.9%

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

10. Taylor expanded in cosTheta_O around 0

    \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(1, v\right), \color{blue}{\frac{6931}{10000}}\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

12. Simplified 99.8%

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

Alternative 4: 39.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;e^{cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_O \cdot cosTheta\_i}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= cosTheta_i -1.99999996490334e-13)
   (exp (* cosTheta_O (/ cosTheta_i v)))
   (/ (* cosTheta_O cosTheta_i) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if (cosTheta_i <= -1.99999996490334e-13f) {
		tmp = expf((cosTheta_O * (cosTheta_i / v)));
	} else {
		tmp = (cosTheta_O * cosTheta_i) / v;
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    real(4) :: tmp
    if (costheta_i <= (-1.99999996490334e-13)) then
        tmp = exp((costheta_o * (costheta_i / v)))
    else
        tmp = (costheta_o * costheta_i) / v
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (cosTheta_i <= Float32(-1.99999996490334e-13))
		tmp = exp(Float32(cosTheta_O * Float32(cosTheta_i / v)));
	else
		tmp = Float32(Float32(cosTheta_O * cosTheta_i) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.0);
	if (cosTheta_i <= single(-1.99999996490334e-13))
		tmp = exp((cosTheta_O * (cosTheta_i / v)));
	else
		tmp = (cosTheta_O * cosTheta_i) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if cosTheta_i < -1.99999996e-13

   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

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 20.2%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 20.2%

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

   if -1.99999996e-13 < cosTheta_i

   1. Initial program 99.4%

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

   3. Taylor expanded in cosTheta_i around inf

      \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

      3. /-lowering-/.f32 11.6%

         \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

   5. Simplified 11.6%

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

   6. Taylor expanded in cosTheta_O around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} + \color{blue}{1} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \color{blue}{1}\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), 1\right) \]

      4. *-lowering-*.f32 6.2%

         \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), 1\right) \]

   8. Simplified 6.2%

      \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} + 1} \]

   9. Taylor expanded in cosTheta_O around inf

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

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

       2. *-lowering-*.f32 42.4%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

   11. Simplified 42.4%

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

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. log-rec N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right)\right), v\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right), v\right) \]

   15. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right), v\right) \]

7. Simplified 99.9%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{6931}{10000} - \frac{1}{v}\right)}\right)\right), v\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right), v\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right), v\right) \]

   5. /-lowering-/.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right), v\right) \]

10. Simplified 99.8%

    \[\leadsto \frac{0.5 \cdot e^{\color{blue}{0.6931 + \frac{-1}{v}}}}{v} \]
11. Step-by-step derivation

    1. associate-/l* N/A

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

    2. clear-num N/A

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

    3. un-div-inv N/A

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

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{v}{e^{\frac{6931}{10000} + \frac{-1}{v}}}\right)}\right) \]

    5. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \color{blue}{\left(e^{\frac{6931}{10000} + \frac{-1}{v}}\right)}\right)\right) \]

    6. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \frac{-1}{v}\right)\right)\right)\right) \]

    7. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right) \]

    8. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right) \]

12. Applied egg-rr 99.8%

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

Alternative 6: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * (Float32(exp(1)) ^ Float32(Float32(-1.0) / v))) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * (single(2.71828182845904523536) ^ (single(-1.0) / v))) / v;
end

Derivation

1. Initial program 99.5%

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

   1. log-rec N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right)\right), v\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right), v\right) \]

   15. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right), v\right) \]

7. Simplified 99.9%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{6931}{10000} - \frac{1}{v}\right)}\right)\right), v\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right), v\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right), v\right) \]

   5. /-lowering-/.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right), v\right) \]

10. Simplified 99.8%

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

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{-1}{v}\right)}\right)\right), v\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 98.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(-1, v\right)\right)\right), v\right) \]

13. Simplified 98.9%

    \[\leadsto \frac{0.5 \cdot e^{\color{blue}{\frac{-1}{v}}}}{v} \]
14. Step-by-step derivation

    1. *-lft-identity N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{1 \cdot \frac{-1}{v}}\right)\right), v\right) \]

    2. exp-prod N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left({\left(e^{1}\right)}^{\left(\frac{-1}{v}\right)}\right)\right), v\right) \]

    3. pow-lowering-pow.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{pow.f32}\left(\left(e^{1}\right), \left(\frac{-1}{v}\right)\right)\right), v\right) \]

    4. exp-1-e N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{-1}{v}\right)\right)\right), v\right) \]

    5. E-lowering-E.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{v}\right)\right)\right), v\right) \]

    6. /-lowering-/.f32 98.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, v\right)\right)\right), v\right) \]

15. Applied egg-rr 98.9%

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

Alternative 7: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. log-rec N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right)\right), v\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right), v\right) \]

   15. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right), v\right) \]

7. Simplified 99.9%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{6931}{10000} - \frac{1}{v}\right)}\right)\right), v\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right), v\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right), v\right) \]

   5. /-lowering-/.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right), v\right) \]

10. Simplified 99.8%

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

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{-1}{v}\right)}\right)\right), v\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 98.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(-1, v\right)\right)\right), v\right) \]

13. Simplified 98.9%

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

Alternative 8: 98.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. log-rec N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(e^{\left(\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v} - \frac{sinTheta\_i \cdot sinTheta\_O}{v}\right) - \frac{1}{v}\right) + \frac{6931}{10000}}\right), \color{blue}{\left(e^{\log \left(2 \cdot v\right)}\right)}\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in sinTheta_i around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right), \color{blue}{v}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(e^{\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}}\right)\right), v\right) \]

   4. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{6931}{10000} + \frac{cosTheta\_O \cdot cosTheta\_i}{v}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \frac{6931}{10000}\right) - \frac{1}{v}\right)\right)\right), v\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v} + \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   10. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} - \frac{1}{v}\right)\right)\right)\right), v\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right)\right), v\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right)\right), v\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right)\right), v\right) \]

   15. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right), \mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right)\right), v\right) \]

7. Simplified 99.9%

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

8. Taylor expanded in cosTheta_O around 0

   \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{6931}{10000} - \frac{1}{v}\right)}\right)\right), v\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\left(\frac{6931}{10000} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)\right)\right), v\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{\mathsf{neg}\left(1\right)}{v}\right)\right)\right)\right), v\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \left(\frac{-1}{v}\right)\right)\right)\right), v\right) \]

   5. /-lowering-/.f32 99.8%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{+.f32}\left(\frac{6931}{10000}, \mathsf{/.f32}\left(-1, v\right)\right)\right)\right), v\right) \]

10. Simplified 99.8%

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

11. Taylor expanded in v around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\color{blue}{\left(\frac{-1}{v}\right)}\right)\right), v\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f32 98.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(-1, v\right)\right)\right), v\right) \]

13. Simplified 98.9%

    \[\leadsto \frac{0.5 \cdot e^{\color{blue}{\frac{-1}{v}}}}{v} \]
14. Step-by-step derivation

    1. associate-/l* N/A

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

    2. clear-num N/A

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

    3. div-inv N/A

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

    4. exp-neg N/A

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

    5. distribute-neg-frac N/A

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

    6. metadata-eval N/A

       \[\leadsto \frac{1}{2} \cdot \frac{1}{v \cdot e^{\frac{1}{v}}} \]

    7. un-div-inv N/A

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

    8. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left(v \cdot e^{\frac{1}{v}}\right)}\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(v, \color{blue}{\left(e^{\frac{1}{v}}\right)}\right)\right) \]

    10. exp-lowering-exp.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(v, \mathsf{exp.f32}\left(\left(\frac{1}{v}\right)\right)\right)\right) \]

    11. /-lowering-/.f32 98.9%

        \[\leadsto \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(v, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(1, v\right)\right)\right)\right) \]

15. Applied egg-rr 98.9%

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

Alternative 9: 37.9% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

   3. /-lowering-/.f32 13.1%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

5. Simplified 13.1%

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

6. Taylor expanded in cosTheta_O around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{cosTheta\_O \cdot cosTheta\_i}{v} + \color{blue}{1} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right), \color{blue}{1}\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), v\right), 1\right) \]

   4. *-lowering-*.f32 6.2%

      \[\leadsto \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right), 1\right) \]

8. Simplified 6.2%

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v} + 1} \]

9. Taylor expanded in cosTheta_O around inf

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

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\left(cosTheta\_O \cdot cosTheta\_i\right), \color{blue}{v}\right) \]

    2. *-lowering-*.f32 36.6%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, cosTheta\_i\right), v\right) \]

11. Simplified 36.6%

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

Alternative 10: 6.4% accurate, 223.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in cosTheta_i around inf

   \[\leadsto \mathsf{exp.f32}\left(\color{blue}{\left(\frac{cosTheta\_O \cdot cosTheta\_i}{v}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \left(\frac{cosTheta\_i}{v}\right)\right)\right) \]

   3. /-lowering-/.f32 13.1%

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(cosTheta\_O, \mathsf{/.f32}\left(cosTheta\_i, v\right)\right)\right) \]

5. Simplified 13.1%

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

6. Taylor expanded in cosTheta_O around 0

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

8. Simplified 6.4%

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

Reproduce

herbie shell --seed 2024164 
(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))))))