HairBSDF, Mp, lower

Percentage Accurate: 99.6% → 99.7%

Time: 15.4s

Alternatives: 9

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	t_0 = Float32(fma(sinTheta_i, Float32(-sinTheta_O), fma(cosTheta_O, cosTheta_i, Float32(-1.0))) / Float32(v * Float32(2.0)))
	return Float32(Float32(exp(Float32(Float32(0.6931) + t_0)) / v) * Float32(exp(t_0) / Float32(2.0)))
end

Derivation

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

4. Applied egg-rr 99.2%

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

   1. lift-neg.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. lift-/.f32 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-exp.f32 99.1

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

6. Applied egg-rr 99.1%

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

8. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

4. Applied egg-rr 99.2%

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

   1. lift-exp.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. div-inv N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lift-/.f32 N/A

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

   9. *-commutative N/A

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

   10. rem-exp-log N/A

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

   11. lift-log.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. prod-exp N/A

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

   14. lift-log.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. log-div N/A

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

   17. associate-+l- N/A

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

   18. exp-diff N/A

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

   19. rem-exp-log N/A

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

   20. lower-/.f32 N/A

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

   21. lower-exp.f32 N/A

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

   22. lower--.f32 N/A

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

   23. lower-log.f32 99.2

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-*.f32 N/A

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

   2. div-exp N/A

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

   3. lower-exp.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-+.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. metadata-eval 99.6

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

9. Simplified 99.6%

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

10. Taylor expanded in cosTheta_O around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower-exp.f32 N/A

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

    4. associate--r+ N/A

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

    5. sub-neg N/A

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

    6. lower-+.f32 N/A

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

    7. lower--.f32 N/A

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

    8. lower-log.f32 N/A

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

    9. distribute-neg-frac N/A

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

    10. metadata-eval N/A

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

    11. lower-/.f32 99.6

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

12. Simplified 99.6%

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

Alternative 3: 99.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (* v (exp (- (- 0.6931) (/ (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 - (fmaf(cosTheta_O, cosTheta_i, -1.0f) / v))));
}

Julia

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

Derivation

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

4. Applied egg-rr 99.6%

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

   1. lift-neg.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. associate-/r* N/A

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

   9. clear-num N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 N/A

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in sinTheta_i around 0

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

   1. lower-/.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. associate--l+ N/A

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

   4. div-sub N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 99.4

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

9. Simplified 99.4%

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

    1. lift-fma.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-+.f32 N/A

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

    4. lift-exp.f32 N/A

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

    5. lift-/.f32 N/A

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

    6. associate-/r/ N/A

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

    7. metadata-eval N/A

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

    8. lift-/.f32 N/A

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

    9. clear-num N/A

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

    10. un-div-inv N/A

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

    11. lower-/.f32 N/A

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

    12. div-inv N/A

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

    13. lower-*.f32 N/A

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

    14. lift-exp.f32 N/A

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

    15. rec-exp N/A

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

    16. lower-exp.f32 N/A

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 5: 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 99.2%

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

3. Taylor expanded in cosTheta_i around 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. lower-*.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. lower-/.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. lower-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. lower-+.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. lower-*.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 99.2%

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

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. lower-/.f32 99.2

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

8. Simplified 99.2%

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

Alternative 6: 99.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 sinTheta_i around 0

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

   1. lower-exp.f32 N/A

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f32 N/A

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

   7. rem-exp-log N/A

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

   8. lower-log.f32 N/A

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

   9. rem-exp-log N/A

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

   10. lower-/.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 99.2

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

5. Simplified 99.2%

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

   1. lift-/.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. lift-log.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. log-div N/A

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

   10. associate-+r- N/A

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

   11. exp-diff N/A

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

   12. rem-exp-log N/A

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

   13. lower-/.f32 N/A

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 98.9

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

10. Simplified 98.9%

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

Alternative 7: 97.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 sinTheta_i around 0

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

   1. lower-exp.f32 N/A

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f32 N/A

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

   7. rem-exp-log N/A

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

   8. lower-log.f32 N/A

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

   9. rem-exp-log N/A

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

   10. lower-/.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 97.0

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

8. Simplified 97.0%

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

9. Taylor expanded in cosTheta_O around 0

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

    1. metadata-eval N/A

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

    2. distribute-neg-frac N/A

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

    3. lower-exp.f32 N/A

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

    4. distribute-neg-frac N/A

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

    5. metadata-eval N/A

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

    6. lower-/.f32 97.0

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

11. Simplified 97.0%

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

    1. clear-num N/A

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

    2. div-inv N/A

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

    3. clear-num N/A

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

    4. lift-/.f32 N/A

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

    5. exp-prod N/A

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

    6. lift-exp.f32 N/A

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

    7. lower-pow.f32 97.0

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

    8. lift-exp.f32 N/A

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

    9. exp-1-e N/A

       \[\leadsto {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{-1}{v}\right)} \]

    10. lower-E.f32 97.0

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

13. Applied egg-rr 97.0%

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

Alternative 8: 97.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 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 = exp(((-1.0e0) / v))
end function

Julia

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

MATLAB

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

Derivation

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 sinTheta_i around 0

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

   1. lower-exp.f32 N/A

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f32 N/A

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

   7. rem-exp-log N/A

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

   8. lower-log.f32 N/A

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

   9. rem-exp-log N/A

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

   10. lower-/.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in v around 0

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

   1. lower-/.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 97.0

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

8. Simplified 97.0%

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

9. Taylor expanded in cosTheta_O around 0

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

    1. metadata-eval N/A

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

    2. distribute-neg-frac N/A

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

    3. lower-exp.f32 N/A

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

    4. distribute-neg-frac N/A

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

    5. metadata-eval N/A

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

    6. lower-/.f32 97.0

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

11. Simplified 97.0%

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

Alternative 9: 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 99.2%

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

3. Taylor expanded in sinTheta_i around 0

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

   1. lower-exp.f32 N/A

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

   2. associate-+r+ N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f32 N/A

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

   7. rem-exp-log N/A

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

   8. lower-log.f32 N/A

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

   9. rem-exp-log N/A

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

   10. lower-/.f32 N/A

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

   11. lower-+.f32 N/A

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

   12. div-sub N/A

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

   13. lower-/.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f32 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in cosTheta_O around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 13.2

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

8. Simplified 13.2%

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

9. Taylor expanded in cosTheta_O around 0

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

11. Simplified 6.5%

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

Reproduce

herbie shell --seed 2024208 
(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))))))