Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 9.9s

Alternatives: 9

Speedup: 21.8×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log1p (* u -4.0)) (- s)))

C

float code(float s, float u) {
	return log1pf((u * -4.0f)) * -s;
}

Julia

function code(s, u)
	return Float32(log1p(Float32(u * Float32(-4.0))) * Float32(-s))
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(1 - 4 \cdot u\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto s \cdot \left(-1 \cdot \color{blue}{\log \left(1 - 4 \cdot u\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(s \cdot -1\right) \cdot \color{blue}{\log \left(1 - 4 \cdot u\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - 4 \cdot u\right) \cdot \color{blue}{\left(s \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - 4 \cdot u\right), \color{blue}{\left(s \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(4 \cdot u\right)\right)\right), \left(s \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(4 \cdot u\right)\right)\right), \left(\color{blue}{s} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(4 \cdot u\right)\right)\right), \left(\color{blue}{s} \cdot -1\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(u \cdot 4\right)\right)\right), \left(s \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(u \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right), \left(s \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(u, \left(\mathsf{neg}\left(4\right)\right)\right)\right), \left(s \cdot -1\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(u, -4\right)\right), \left(s \cdot -1\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(u, -4\right)\right), \left(-1 \cdot \color{blue}{s}\right)\right) \]

   14. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(u, -4\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right) \]

   15. neg-lowering-neg.f32 99.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(u, -4\right)\right), \mathsf{neg.f32}\left(s\right)\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 94.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot \left(4 + u \cdot 8\right) + \frac{s \cdot 455.1111111111111}{21.333333333333332 + u \cdot -64} \cdot \left(u \cdot u\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  u
  (+
   (* s (+ 4.0 (* u 8.0)))
   (*
    (/ (* s 455.1111111111111) (+ 21.333333333333332 (* u -64.0)))
    (* u u)))))

C

float code(float s, float u) {
	return u * ((s * (4.0f + (u * 8.0f))) + (((s * 455.1111111111111f) / (21.333333333333332f + (u * -64.0f))) * (u * u)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * (4.0e0 + (u * 8.0e0))) + (((s * 455.1111111111111e0) / (21.333333333333332e0 + (u * (-64.0e0)))) * (u * u)))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(s * Float32(Float32(4.0) + Float32(u * Float32(8.0)))) + Float32(Float32(Float32(s * Float32(455.1111111111111)) / Float32(Float32(21.333333333333332) + Float32(u * Float32(-64.0)))) * Float32(u * u))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * (single(4.0) + (u * single(8.0)))) + (((s * single(455.1111111111111)) / (single(21.333333333333332) + (u * single(-64.0)))) * (u * u)));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{\left(4 \cdot s + u \cdot \left(8 \cdot s + u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)\right)}\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(4 \cdot s + \left(\left(8 \cdot s\right) \cdot u + \color{blue}{\left(u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right) \cdot u}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(4 \cdot s + \left(8 \cdot \left(s \cdot u\right) + \color{blue}{\left(u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)} \cdot u\right)\right)\right) \]

   4. associate-+r+ N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(\left(4 \cdot s + 8 \cdot \left(s \cdot u\right)\right) + \color{blue}{\left(u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right) \cdot u}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(4 \cdot s + 8 \cdot \left(s \cdot u\right)\right), \color{blue}{\left(\left(u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right) \cdot u\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(s \cdot 4 + 8 \cdot \left(s \cdot u\right)\right), \left(\left(\color{blue}{u} \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right) \cdot u\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(s \cdot 4 + \left(s \cdot u\right) \cdot 8\right), \left(\left(u \cdot \color{blue}{\left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)}\right) \cdot u\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(s \cdot 4 + s \cdot \left(u \cdot 8\right)\right), \left(\left(u \cdot \color{blue}{\left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)}\right) \cdot u\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(s \cdot 4 + s \cdot \left(8 \cdot u\right)\right), \left(\left(u \cdot \left(\frac{64}{3} \cdot s + \color{blue}{64 \cdot \left(s \cdot u\right)}\right)\right) \cdot u\right)\right)\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(s \cdot \left(4 + 8 \cdot u\right)\right), \left(\color{blue}{\left(u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)} \cdot u\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \left(4 + 8 \cdot u\right)\right), \left(\color{blue}{\left(u \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right)} \cdot u\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \left(8 \cdot u\right)\right)\right), \left(\left(u \cdot \color{blue}{\left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)}\right) \cdot u\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \left(u \cdot 8\right)\right)\right), \left(\left(u \cdot \left(\frac{64}{3} \cdot s + \color{blue}{64 \cdot \left(s \cdot u\right)}\right)\right) \cdot u\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \left(\left(u \cdot \left(\frac{64}{3} \cdot s + \color{blue}{64 \cdot \left(s \cdot u\right)}\right)\right) \cdot u\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \left(\left(\left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right) \cdot u\right) \cdot u\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \left(\left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right) \cdot \color{blue}{\left(u \cdot u\right)}\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \left(\left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right) \cdot {u}^{\color{blue}{2}}\right)\right)\right) \]

5. Simplified 93.5%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(4 + u \cdot 8\right) + \left(s \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\left(s \cdot \frac{64}{3} + s \cdot \left(u \cdot 64\right)\right), \mathsf{*.f32}\left(\color{blue}{u}, u\right)\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{\left(s \cdot \frac{64}{3}\right) \cdot \left(s \cdot \frac{64}{3}\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)}{s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)}\right), \mathsf{*.f32}\left(\color{blue}{u}, u\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\left(s \cdot \frac{64}{3}\right) \cdot \left(s \cdot \frac{64}{3}\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{u}, u\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\left(\frac{64}{3} \cdot s\right) \cdot \left(s \cdot \frac{64}{3}\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\left(\frac{64}{3} \cdot s\right) \cdot \left(\frac{64}{3} \cdot s\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\left(\frac{64}{3} \cdot s\right) \cdot \left(\frac{64}{3} \cdot s\right)\right), \left(\left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(\frac{64}{3} \cdot s\right), \left(\frac{64}{3} \cdot s\right)\right), \left(\left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(s \cdot \frac{64}{3}\right), \left(\frac{64}{3} \cdot s\right)\right), \left(\left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \left(\frac{64}{3} \cdot s\right)\right), \left(\left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \left(s \cdot \frac{64}{3}\right)\right), \left(\left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \left(\left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \mathsf{*.f32}\left(\left(s \cdot \left(u \cdot 64\right)\right), \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left(u \cdot 64\right)\right), \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right), \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right), \mathsf{*.f32}\left(s, \left(u \cdot 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right), \left(s \cdot \frac{64}{3} - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right), \left(\frac{64}{3} \cdot s - s \cdot \left(u \cdot 64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \frac{64}{3}\right), \mathsf{*.f32}\left(s, \frac{64}{3}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right), \mathsf{\_.f32}\left(\left(\frac{64}{3} \cdot s\right), \left(s \cdot \left(u \cdot 64\right)\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

7. Applied egg-rr 91.8%

   \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot 8\right) + \color{blue}{\frac{\left(s \cdot 21.333333333333332\right) \cdot \left(s \cdot 21.333333333333332\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)}{s \cdot 21.333333333333332 - s \cdot \left(u \cdot 64\right)}} \cdot \left(u \cdot u\right)\right) \]
8. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{\left(s \cdot \frac{64}{3}\right) \cdot \left(s \cdot \frac{64}{3}\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)}{s \cdot \left(\frac{64}{3} - u \cdot 64\right)}\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\left(\frac{\frac{\left(s \cdot \frac{64}{3}\right) \cdot \left(s \cdot \frac{64}{3}\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)}{s}}{\frac{64}{3} - u \cdot 64}\right), \mathsf{*.f32}\left(\color{blue}{u}, u\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(\frac{\left(s \cdot \frac{64}{3}\right) \cdot \left(s \cdot \frac{64}{3}\right) - \left(s \cdot \left(u \cdot 64\right)\right) \cdot \left(s \cdot \left(u \cdot 64\right)\right)}{s}\right), \left(\frac{64}{3} - u \cdot 64\right)\right), \mathsf{*.f32}\left(\color{blue}{u}, u\right)\right)\right)\right) \]

9. Applied egg-rr 91.7%

   \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot 8\right) + \color{blue}{\frac{\frac{\left(s \cdot s\right) \cdot \left(455.1111111111111 - \left(u \cdot u\right) \cdot 4096\right)}{s}}{21.333333333333332 + u \cdot -64}} \cdot \left(u \cdot u\right)\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(\frac{4096}{9} \cdot s\right)}, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, -64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(s \cdot \frac{4096}{9}\right), \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, -64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

    2. *-lowering-*.f32 94.8%

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, 8\right)\right)\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \frac{4096}{9}\right), \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, -64\right)\right)\right), \mathsf{*.f32}\left(u, u\right)\right)\right)\right) \]

12. Simplified 94.8%

    \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot 8\right) + \frac{\color{blue}{s \cdot 455.1111111111111}}{21.333333333333332 + u \cdot -64} \cdot \left(u \cdot u\right)\right) \]
13. Add Preprocessing

Alternative 3: 93.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) + u \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (+ (* (* u u) (+ 8.0 (* u (+ 21.333333333333332 (* u 64.0))))) (* u 4.0))))

C

float code(float s, float u) {
	return s * (((u * u) * (8.0f + (u * (21.333333333333332f + (u * 64.0f))))) + (u * 4.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (((u * u) * (8.0e0 + (u * (21.333333333333332e0 + (u * 64.0e0))))) + (u * 4.0e0))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(u * u) * Float32(Float32(8.0) + Float32(u * Float32(Float32(21.333333333333332) + Float32(u * Float32(64.0)))))) + Float32(u * Float32(4.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (((u * u) * (single(8.0) + (u * (single(21.333333333333332) + (u * single(64.0)))))) + (u * single(4.0)));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \color{blue}{\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \color{blue}{\left(u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \color{blue}{\left(\frac{64}{3} + 64 \cdot u\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \color{blue}{\left(64 \cdot u\right)}\right)\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \left(u \cdot \color{blue}{64}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 93.7%

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, \color{blue}{64}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 93.7%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(s, \left(u \cdot \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) + \color{blue}{4}\right)\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f32}\left(s, \left(u \cdot \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) + \color{blue}{u \cdot 4}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(s, \left(u \cdot \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) + 4 \cdot \color{blue}{u}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(u \cdot \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)\right), \color{blue}{\left(4 \cdot u\right)}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right), \left(\color{blue}{4} \cdot u\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(u \cdot u\right), \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right), \left(\color{blue}{4} \cdot u\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right), \left(4 \cdot u\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{+.f32}\left(8, \left(u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)\right), \left(4 \cdot u\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \left(\frac{64}{3} + u \cdot 64\right)\right)\right)\right), \left(4 \cdot u\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \left(u \cdot 64\right)\right)\right)\right)\right), \left(4 \cdot u\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right)\right), \left(4 \cdot u\right)\right)\right) \]

   12. *-lowering-*.f32 93.9%

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right)\right), \mathsf{*.f32}\left(4, \color{blue}{u}\right)\right)\right) \]

7. Applied egg-rr 93.9%

   \[\leadsto s \cdot \color{blue}{\left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) + 4 \cdot u\right)} \]

8. Final simplification 93.9%

   \[\leadsto s \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) + u \cdot 4\right) \]
9. Add Preprocessing

Alternative 4: 93.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (+ 4.0 (* u (+ 8.0 (* u (+ 21.333333333333332 (* u 64.0)))))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * (8.0f + (u * (21.333333333333332f + (u * 64.0f)))))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * (8.0e0 + (u * (21.333333333333332e0 + (u * 64.0e0)))))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(Float32(8.0) + Float32(u * Float32(Float32(21.333333333333332) + Float32(u * Float32(64.0)))))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * (single(8.0) + (u * (single(21.333333333333332) + (u * single(64.0))))))));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \color{blue}{\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \color{blue}{\left(u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \color{blue}{\left(\frac{64}{3} + 64 \cdot u\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \color{blue}{\left(64 \cdot u\right)}\right)\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \left(u \cdot \color{blue}{64}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 93.7%

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, \color{blue}{64}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 93.7%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 5: 91.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 4 + \left(u \cdot s\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* u (+ (* s 4.0) (* (* u s) (+ 8.0 (* u 21.333333333333332))))))

C

float code(float s, float u) {
	return u * ((s * 4.0f) + ((u * s) * (8.0f + (u * 21.333333333333332f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * 4.0e0) + ((u * s) * (8.0e0 + (u * 21.333333333333332e0))))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(s * Float32(4.0)) + Float32(Float32(u * s) * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332))))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * single(4.0)) + ((u * s) * (single(8.0) + (u * single(21.333333333333332)))));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + \frac{64}{3} \cdot \left(s \cdot u\right)\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{\left(4 \cdot s + u \cdot \left(8 \cdot s + \frac{64}{3} \cdot \left(s \cdot u\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(4 \cdot s\right), \color{blue}{\left(u \cdot \left(8 \cdot s + \frac{64}{3} \cdot \left(s \cdot u\right)\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(s \cdot 4\right), \left(\color{blue}{u} \cdot \left(8 \cdot s + \frac{64}{3} \cdot \left(s \cdot u\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \left(\color{blue}{u} \cdot \left(8 \cdot s + \frac{64}{3} \cdot \left(s \cdot u\right)\right)\right)\right)\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \left(\left(8 \cdot s\right) \cdot u + \color{blue}{\left(\frac{64}{3} \cdot \left(s \cdot u\right)\right) \cdot u}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \left(8 \cdot \left(s \cdot u\right) + \color{blue}{\left(\frac{64}{3} \cdot \left(s \cdot u\right)\right)} \cdot u\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \left(\left(s \cdot u\right) \cdot 8 + \color{blue}{\left(\frac{64}{3} \cdot \left(s \cdot u\right)\right)} \cdot u\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \left(\left(s \cdot u\right) \cdot 8 + \left(\left(s \cdot u\right) \cdot \frac{64}{3}\right) \cdot u\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \left(\left(s \cdot u\right) \cdot 8 + \left(s \cdot u\right) \cdot \color{blue}{\left(\frac{64}{3} \cdot u\right)}\right)\right)\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \left(\left(s \cdot u\right) \cdot \color{blue}{\left(8 + \frac{64}{3} \cdot u\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{\left(8 + \frac{64}{3} \cdot u\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), \left(\color{blue}{8} + \frac{64}{3} \cdot u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), \mathsf{+.f32}\left(8, \color{blue}{\left(\frac{64}{3} \cdot u\right)}\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), \mathsf{+.f32}\left(8, \left(u \cdot \color{blue}{\frac{64}{3}}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 92.2%

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, 4\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \color{blue}{\frac{64}{3}}\right)\right)\right)\right)\right) \]

5. Simplified 92.2%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot 4 + \left(s \cdot u\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right)} \]

6. Final simplification 92.2%

   \[\leadsto u \cdot \left(s \cdot 4 + \left(u \cdot s\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right) \]
7. Add Preprocessing

Alternative 6: 90.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (+ 4.0 (* u (+ 8.0 (* u 21.333333333333332)))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * (8.0f + (u * 21.333333333333332f)))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * (8.0e0 + (u * 21.333333333333332e0)))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332)))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * (single(8.0) + (u * single(21.333333333333332))))));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \color{blue}{\left(4 + u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \color{blue}{\left(u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \color{blue}{\left(8 + \frac{64}{3} \cdot u\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \color{blue}{\left(\frac{64}{3} \cdot u\right)}\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \left(u \cdot \color{blue}{\frac{64}{3}}\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 92.0%

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \color{blue}{\frac{64}{3}}\right)\right)\right)\right)\right)\right) \]

5. Simplified 92.0%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right)} \]
6. Add Preprocessing

Alternative 7: 86.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 4 + s \cdot \left(u \cdot 8\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (+ (* s 4.0) (* s (* u 8.0)))))

C

float code(float s, float u) {
	return u * ((s * 4.0f) + (s * (u * 8.0f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * 4.0e0) + (s * (u * 8.0e0)))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(s * Float32(4.0)) + Float32(s * Float32(u * Float32(8.0)))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * single(4.0)) + (s * (u * single(8.0))));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + 8 \cdot \left(s \cdot u\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{\left(4 \cdot s + 8 \cdot \left(s \cdot u\right)\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(s \cdot 4 + \color{blue}{8} \cdot \left(s \cdot u\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(s \cdot 4 + \left(s \cdot u\right) \cdot \color{blue}{8}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(s \cdot 4 + s \cdot \color{blue}{\left(u \cdot 8\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(s \cdot 4 + s \cdot \left(8 \cdot \color{blue}{u}\right)\right)\right) \]

   6. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(s \cdot \color{blue}{\left(4 + 8 \cdot u\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + 8 \cdot u\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \color{blue}{\left(8 \cdot u\right)}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \left(u \cdot \color{blue}{8}\right)\right)\right)\right) \]

   10. *-lowering-*.f32 88.1%

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \color{blue}{8}\right)\right)\right)\right) \]

5. Simplified 88.1%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(s \cdot \left(u \cdot 8 + \color{blue}{4}\right)\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(s \cdot \left(u \cdot 8\right) + \color{blue}{s \cdot 4}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(s \cdot \left(u \cdot 8\right)\right), \color{blue}{\left(s \cdot 4\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \left(u \cdot 8\right)\right), \left(\color{blue}{s} \cdot 4\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 8\right)\right), \left(s \cdot 4\right)\right)\right) \]

   6. *-lowering-*.f32 88.3%

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, 8\right)\right), \mathsf{*.f32}\left(s, \color{blue}{4}\right)\right)\right) \]

7. Applied egg-rr 88.3%

   \[\leadsto u \cdot \color{blue}{\left(s \cdot \left(u \cdot 8\right) + s \cdot 4\right)} \]

8. Final simplification 88.3%

   \[\leadsto u \cdot \left(s \cdot 4 + s \cdot \left(u \cdot 8\right)\right) \]
9. Add Preprocessing

Alternative 8: 86.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* u (+ 4.0 (* u 8.0)))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * 8.0f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * 8.0e0)))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(8.0)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * single(8.0))));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(u \cdot \left(4 + 8 \cdot u\right)\right)}\right) \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \color{blue}{\left(4 + 8 \cdot u\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \color{blue}{\left(8 \cdot u\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \left(u \cdot \color{blue}{8}\right)\right)\right)\right) \]

   4. *-lowering-*.f32 88.1%

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \color{blue}{8}\right)\right)\right)\right) \]

5. Simplified 88.1%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot 8\right)\right)} \]
6. Add Preprocessing

Alternative 9: 73.7% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* u 4.0)))

C

float code(float s, float u) {
	return s * (u * 4.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * 4.0e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(4.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * single(4.0));
end

Derivation

1. Initial program 61.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

   2. associate-*l* N/A

      \[\leadsto s \cdot \color{blue}{\left(u \cdot 4\right)} \]

   3. *-commutative N/A

      \[\leadsto s \cdot \left(4 \cdot \color{blue}{u}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(4 \cdot u\right)}\right) \]

   5. *-lowering-*.f32 75.7%

      \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(4, \color{blue}{u}\right)\right) \]

5. Simplified 75.7%

   \[\leadsto \color{blue}{s \cdot \left(4 \cdot u\right)} \]

6. Final simplification 75.7%

   \[\leadsto s \cdot \left(u \cdot 4\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))