HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.5%

Time: 16.4s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(v \cdot 2\right) \cdot e^{-0.6931}\right)}^{-0.5}\\ \left(t\_0 \cdot t\_0\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, -sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)\right)}{v}\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (pow (* (* v 2.0) (exp -0.6931)) -0.5)))
   (*
    (* t_0 t_0)
    (pow
     E
     (/ (fma sinTheta_O (- sinTheta_i) (fma cosTheta_i cosTheta_O -1.0)) v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = powf(((v * 2.0f) * expf(-0.6931f)), -0.5f);
	return (t_0 * t_0) * powf(((float) M_E), (fmaf(sinTheta_O, -sinTheta_i, fmaf(cosTheta_i, cosTheta_O, -1.0f)) / v));
}

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(v * Float32(2.0)) * exp(Float32(-0.6931))) ^ Float32(-0.5)
	return Float32(Float32(t_0 * t_0) * (Float32(exp(1)) ^ Float32(fma(sinTheta_O, Float32(-sinTheta_i), fma(cosTheta_i, cosTheta_O, Float32(-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. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. exp-sum N/A

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

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

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

   5. log-rec N/A

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

   6. unsub-neg N/A

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

   7. exp-diff N/A

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

   8. rem-exp-log N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. sub-div N/A

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

4. Applied egg-rr 99.5%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. exp-prod N/A

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

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

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

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

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

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

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

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

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

   11. neg-lowering-neg.f32 N/A

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

   12. accelerator-lowering-fma.f32 99.5

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

6. Applied egg-rr 99.5%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. sqr-pow N/A

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

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

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

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

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

   6. div-inv N/A

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

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

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

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

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

   9. rec-exp N/A

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

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

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. div-inv N/A

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

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

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

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

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

   17. rec-exp N/A

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

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

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

   19. metadata-eval N/A

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

   20. metadata-eval 99.5

       \[\leadsto \left({\left(\left(v \cdot 2\right) \cdot e^{-0.6931}\right)}^{-0.5} \cdot {\left(\left(v \cdot 2\right) \cdot e^{-0.6931}\right)}^{\color{blue}{-0.5}}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, -sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)\right)}{v}\right)} \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\left({\left(\left(v \cdot 2\right) \cdot e^{-0.6931}\right)}^{-0.5} \cdot {\left(\left(v \cdot 2\right) \cdot e^{-0.6931}\right)}^{-0.5}\right)} \cdot {\left(e^{1}\right)}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, -sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)\right)}{v}\right)} \]

9. Final simplification 99.5%

   \[\leadsto \left({\left(\left(v \cdot 2\right) \cdot e^{-0.6931}\right)}^{-0.5} \cdot {\left(\left(v \cdot 2\right) \cdot e^{-0.6931}\right)}^{-0.5}\right) \cdot {e}^{\left(\frac{\mathsf{fma}\left(sinTheta\_O, -sinTheta\_i, \mathsf{fma}\left(cosTheta\_i, cosTheta\_O, -1\right)\right)}{v}\right)} \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(0.5) / v) * exp(Float32(0.6931))) * (Float32(exp(1)) ^ Float32(fma(cosTheta_O, cosTheta_i, Float32(-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. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. exp-sum N/A

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

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

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

   5. log-rec N/A

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

   6. unsub-neg N/A

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

   7. exp-diff N/A

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

   8. rem-exp-log N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. sub-div N/A

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in sinTheta_O around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. accelerator-lowering-fma.f32 99.5

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

7. Simplified 99.5%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

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

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

   4. exp-1-e N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f32 99.5

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

9. Applied egg-rr 99.5%

   \[\leadsto \frac{e^{0.6931}}{v \cdot 2} \cdot \color{blue}{{e}^{\left(\frac{\mathsf{fma}\left(cosTheta\_O, cosTheta\_i, -1\right)}{v}\right)}} \]
10. Step-by-step derivation

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

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

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

    4. *-commutative N/A

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

    5. associate-/r* N/A

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

    6. metadata-eval N/A

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

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

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

    8. exp-lowering-exp.f32 99.5

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

11. Applied egg-rr 99.5%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(0.6931)) / Float32(v * Float32(2.0))) * (Float32(exp(1)) ^ Float32(Float32(-1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(single(0.6931)) / (v * single(2.0))) * (single(2.71828182845904523536) ^ (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. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. exp-sum N/A

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

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

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

   5. log-rec N/A

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

   6. unsub-neg N/A

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

   7. exp-diff N/A

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

   8. rem-exp-log N/A

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

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

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. sub-div N/A

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in sinTheta_O around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. accelerator-lowering-fma.f32 99.5

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

7. Simplified 99.5%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

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

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

   4. exp-1-e N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f32 99.5

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

9. Applied egg-rr 99.5%

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

10. Taylor expanded in cosTheta_O around 0

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

    1. /-lowering-/.f32 99.5

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

12. Simplified 99.5%

    \[\leadsto \frac{e^{0.6931}}{v \cdot 2} \cdot {e}^{\color{blue}{\left(\frac{-1}{v}\right)}} \]
13. Add Preprocessing

Alternative 4: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

3. Taylor expanded in cosTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

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

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

   8. sub-neg N/A

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

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

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

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

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

5. Simplified 99.5%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. /-lowering-/.f32 99.5

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

8. Simplified 99.5%

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

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

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

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

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

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

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

   6. exp-lowering-exp.f32 99.5

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

10. Applied egg-rr 99.5%

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

11. Final simplification 99.5%

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

Alternative 5: 52.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{elif}\;cosTheta\_i \cdot cosTheta\_O \leq 2.000000036005019 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{-cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_i cosTheta_O) v)))
   (if (<= (* cosTheta_i cosTheta_O) -2.9999999105145657e-35)
     (exp t_0)
     (if (<= (* cosTheta_i cosTheta_O) 2.000000036005019e-35)
       t_0
       (exp (- (* cosTheta_O (/ cosTheta_i v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) / v;
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -2.9999999105145657e-35f) {
		tmp = expf(t_0);
	} else if ((cosTheta_i * cosTheta_O) <= 2.000000036005019e-35f) {
		tmp = t_0;
	} else {
		tmp = expf(-(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) :: t_0
    real(4) :: tmp
    t_0 = (costheta_i * costheta_o) / v
    if ((costheta_i * costheta_o) <= (-2.9999999105145657e-35)) then
        tmp = exp(t_0)
    else if ((costheta_i * costheta_o) <= 2.000000036005019e-35) then
        tmp = t_0
    else
        tmp = exp(-(costheta_o * (costheta_i / v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) / v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-2.9999999105145657e-35))
		tmp = exp(t_0);
	elseif (Float32(cosTheta_i * cosTheta_O) <= Float32(2.000000036005019e-35))
		tmp = t_0;
	else
		tmp = exp(Float32(-Float32(cosTheta_O * Float32(cosTheta_i / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_i * cosTheta_O) / v;
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-2.9999999105145657e-35))
		tmp = exp(t_0);
	elseif ((cosTheta_i * cosTheta_O) <= single(2.000000036005019e-35))
		tmp = t_0;
	else
		tmp = exp(-(cosTheta_O * (cosTheta_i / v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -2.99999991e-35

   1. Initial program 99.2%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 39.3

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

   5. Simplified 39.3%

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. *-lowering-*.f32 39.3

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

   7. Applied egg-rr 39.3%

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

   if -2.99999991e-35 < (*.f32 cosTheta_i cosTheta_O) < 2.00000004e-35

   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

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 6.4

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

   5. Simplified 6.4%

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

   6. Taylor expanded in cosTheta_i around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f32 6.4

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

   8. Simplified 6.4%

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

   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

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

       2. *-lowering-*.f32 61.9

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

   11. Simplified 61.9%

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

   if 2.00000004e-35 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.1%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 5.7

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

   5. Simplified 5.7%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. inv-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. sqr-neg N/A

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

      9. pow2 N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. inv-pow N/A

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

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

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

      15. neg-lowering-neg.f32 N/A

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

      16. /-lowering-/.f32 35.7

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

   7. Applied egg-rr 35.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. un-div-inv N/A

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

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

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

      6. neg-lowering-neg.f32 35.7

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

   9. Applied egg-rr 35.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{elif}\;cosTheta\_i \cdot cosTheta\_O \leq 2.000000036005019 \cdot 10^{-35}:\\ \;\;\;\;\frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{-cosTheta\_O \cdot \frac{cosTheta\_i}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{elif}\;cosTheta\_i \cdot cosTheta\_O \leq 2.000000036005019 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_i \cdot \frac{cosTheta\_O}{-v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_i cosTheta_O) v)))
   (if (<= (* cosTheta_i cosTheta_O) -2.9999999105145657e-35)
     (exp t_0)
     (if (<= (* cosTheta_i cosTheta_O) 2.000000036005019e-35)
       t_0
       (exp (* cosTheta_i (/ cosTheta_O (- v))))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) / v;
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -2.9999999105145657e-35f) {
		tmp = expf(t_0);
	} else if ((cosTheta_i * cosTheta_O) <= 2.000000036005019e-35f) {
		tmp = t_0;
	} else {
		tmp = expf((cosTheta_i * (cosTheta_O / -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) :: t_0
    real(4) :: tmp
    t_0 = (costheta_i * costheta_o) / v
    if ((costheta_i * costheta_o) <= (-2.9999999105145657e-35)) then
        tmp = exp(t_0)
    else if ((costheta_i * costheta_o) <= 2.000000036005019e-35) then
        tmp = t_0
    else
        tmp = exp((costheta_i * (costheta_o / -v)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) / v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-2.9999999105145657e-35))
		tmp = exp(t_0);
	elseif (Float32(cosTheta_i * cosTheta_O) <= Float32(2.000000036005019e-35))
		tmp = t_0;
	else
		tmp = exp(Float32(cosTheta_i * Float32(cosTheta_O / Float32(-v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_i * cosTheta_O) / v;
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-2.9999999105145657e-35))
		tmp = exp(t_0);
	elseif ((cosTheta_i * cosTheta_O) <= single(2.000000036005019e-35))
		tmp = t_0;
	else
		tmp = exp((cosTheta_i * (cosTheta_O / -v)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -2.99999991e-35

   1. Initial program 99.2%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 39.3

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

   5. Simplified 39.3%

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. *-lowering-*.f32 39.3

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

   7. Applied egg-rr 39.3%

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

   if -2.99999991e-35 < (*.f32 cosTheta_i cosTheta_O) < 2.00000004e-35

   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

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 6.4

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

   5. Simplified 6.4%

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

   6. Taylor expanded in cosTheta_i around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f32 6.4

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

   8. Simplified 6.4%

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

   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

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

       2. *-lowering-*.f32 61.9

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

   11. Simplified 61.9%

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

   if 2.00000004e-35 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.1%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 5.7

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

   5. Simplified 5.7%

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

      1. frac-2neg N/A

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

      2. neg-mul-1 N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow-pow N/A

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

      13. pow2 N/A

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

      14. sqr-neg N/A

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

      15. pow2 N/A

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

      16. pow-pow N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. inv-pow N/A

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

      20. clear-num N/A

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

      21. /-rgt-identity N/A

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

      22. /-lowering-/.f32 N/A

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

      23. neg-lowering-neg.f32 35.7

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

   7. Applied egg-rr 35.7%

      \[\leadsto e^{cosTheta\_i \cdot \color{blue}{\frac{-cosTheta\_O}{v}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{elif}\;cosTheta\_i \cdot cosTheta\_O \leq 2.000000036005019 \cdot 10^{-35}:\\ \;\;\;\;\frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{else}:\\ \;\;\;\;e^{cosTheta\_i \cdot \frac{cosTheta\_O}{-v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 2.1× speedup

Math

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

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(Float32(0.5) / 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. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

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

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

   8. sub-neg N/A

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

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

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

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

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

5. Simplified 99.5%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. /-lowering-/.f32 99.5

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

8. Simplified 99.5%

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

Alternative 8: 45.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (/ (* cosTheta_i cosTheta_O) v)))
   (if (<= (* cosTheta_i cosTheta_O) -2.9999999105145657e-35) (exp t_0) t_0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = (cosTheta_i * cosTheta_O) / v;
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -2.9999999105145657e-35f) {
		tmp = expf(t_0);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = (costheta_i * costheta_o) / v
    if ((costheta_i * costheta_o) <= (-2.9999999105145657e-35)) then
        tmp = exp(t_0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(Float32(cosTheta_i * cosTheta_O) / v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-2.9999999105145657e-35))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = (cosTheta_i * cosTheta_O) / v;
	tmp = single(0.0);
	if ((cosTheta_i * cosTheta_O) <= single(-2.9999999105145657e-35))
		tmp = exp(t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -2.99999991e-35

   1. Initial program 99.2%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 39.3

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

   5. Simplified 39.3%

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

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

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

      2. associate-*r/ N/A

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

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

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

      4. *-lowering-*.f32 39.3

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

   7. Applied egg-rr 39.3%

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

   if -2.99999991e-35 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.6%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 6.2

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

   5. Simplified 6.2%

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

   6. Taylor expanded in cosTheta_i around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f32 6.4

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

   8. Simplified 6.4%

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

   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

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

       2. *-lowering-*.f32 46.9

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

   11. Simplified 46.9%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (if (<= (* cosTheta_i cosTheta_O) -2.9999999105145657e-35)
   (exp (* cosTheta_i (/ cosTheta_O v)))
   (/ (* cosTheta_i cosTheta_O) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float tmp;
	if ((cosTheta_i * cosTheta_O) <= -2.9999999105145657e-35f) {
		tmp = expf((cosTheta_i * (cosTheta_O / v)));
	} else {
		tmp = (cosTheta_i * cosTheta_O) / 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 * costheta_o) <= (-2.9999999105145657e-35)) then
        tmp = exp((costheta_i * (costheta_o / v)))
    else
        tmp = (costheta_i * costheta_o) / v
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = Float32(0.0)
	if (Float32(cosTheta_i * cosTheta_O) <= Float32(-2.9999999105145657e-35))
		tmp = exp(Float32(cosTheta_i * Float32(cosTheta_O / v)));
	else
		tmp = Float32(Float32(cosTheta_i * cosTheta_O) / 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 * cosTheta_O) <= single(-2.9999999105145657e-35))
		tmp = exp((cosTheta_i * (cosTheta_O / v)));
	else
		tmp = (cosTheta_i * cosTheta_O) / v;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 cosTheta_i cosTheta_O) < -2.99999991e-35

   1. Initial program 99.2%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 39.3

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

   5. Simplified 39.3%

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

   if -2.99999991e-35 < (*.f32 cosTheta_i cosTheta_O)

   1. Initial program 99.6%

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

   3. Taylor expanded in cosTheta_i around inf

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

      4. /-lowering-/.f32 6.2

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

   5. Simplified 6.2%

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

   6. Taylor expanded in cosTheta_i around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f32 6.4

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

   8. Simplified 6.4%

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

   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

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

       2. *-lowering-*.f32 46.9

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

   11. Simplified 46.9%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;cosTheta\_i \cdot cosTheta\_O \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;e^{cosTheta\_i \cdot \frac{cosTheta\_O}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{cosTheta\_i \cdot cosTheta\_O}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.0% 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(Float32(0.5) / 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. Add Preprocessing

3. Taylor expanded in cosTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. exp-sum N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

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

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

   8. sub-neg N/A

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

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

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. distribute-rgt1-in N/A

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

   18. +-commutative N/A

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

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

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

5. Simplified 99.5%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. /-lowering-/.f32 99.5

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

8. Simplified 99.5%

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

9. Taylor expanded in v around 0

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

    1. /-lowering-/.f32 96.6

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

11. Simplified 96.6%

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

Alternative 11: 98.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return exp(Float32(fma(cosTheta_O, cosTheta_i, fma(sinTheta_O, Float32(-sinTheta_i), Float32(-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. Add Preprocessing

3. Taylor expanded in v around 0

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

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

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

   2. sub-neg N/A

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

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

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

   4. +-commutative N/A

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

   5. distribute-neg-in N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. mul-1-neg N/A

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

   8. metadata-eval N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. mul-1-neg N/A

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

   11. neg-lowering-neg.f32 96.3

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

5. Simplified 96.3%

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

Alternative 12: 38.9% accurate, 16.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) / 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_i * costheta_o) / v
end function

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

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

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

   4. /-lowering-/.f32 12.5

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

5. Simplified 12.5%

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

6. Taylor expanded in cosTheta_i around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f32 6.4

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

8. Simplified 6.4%

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

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

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

    2. *-lowering-*.f32 39.2

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

11. Simplified 39.2%

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

12. Final simplification 39.2%

    \[\leadsto \frac{cosTheta\_i \cdot cosTheta\_O}{v} \]
13. Add Preprocessing

Alternative 13: 6.4% accurate, 272.0× 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

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

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

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

   4. /-lowering-/.f32 12.5

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

5. Simplified 12.5%

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

6. Taylor expanded in cosTheta_i around 0

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

8. Simplified 6.5%

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

Reproduce

herbie shell --seed 2024204 
(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))))))