Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.7% → 99.3%

Time: 11.2s

Alternatives: 14

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: 60.7% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right) \cdot \left(-s\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma (log1p (* 4.0 u)) s (* (log1p (* u (* u -16.0))) (- s))))

C

float code(float s, float u) {
	return fmaf(log1pf((4.0f * u)), s, (log1pf((u * (u * -16.0f))) * -s));
}

Julia

function code(s, u)
	return fma(log1p(Float32(Float32(4.0) * u)), s, Float32(log1p(Float32(u * Float32(u * Float32(-16.0)))) * Float32(-s)))
end

Derivation

1. Initial program 58.1%

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

4. Applied egg-rr 99.2%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right)\right)} \]
5. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto s \cdot \left(\log \left(1 + 4 \cdot u\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right)}\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \log \left(1 + 4 \cdot u\right) \cdot s + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right) \cdot s} \]

   3. fma-define N/A

      \[\leadsto \mathsf{fma}\left(\log \left(1 + 4 \cdot u\right), \color{blue}{s}, \left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right) \cdot s\right) \]

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

      \[\leadsto \mathsf{fma.f32}\left(\log \left(1 + 4 \cdot u\right), \color{blue}{s}, \left(\left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right) \cdot s\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{fma.f32}\left(\left(\mathsf{log1p}\left(4 \cdot u\right)\right), s, \left(\left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right) \cdot s\right)\right) \]

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

      \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\left(4 \cdot u\right)\right), s, \left(\left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right) \cdot s\right)\right) \]

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

      \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \left(\left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right) \cdot s\right)\right) \]

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

      \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right)\right), s\right)\right) \]

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

      \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + \left(u \cdot u\right) \cdot -16\right)\right), s\right)\right) \]

   10. log1p-define N/A

       \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right)\right)\right), s\right)\right) \]

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

       \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(\left(u \cdot u\right) \cdot -16\right)\right)\right), s\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(u \cdot \left(u \cdot -16\right)\right)\right)\right), s\right)\right) \]

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

       \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot -16\right)\right)\right)\right), s\right)\right) \]

   14. *-lowering-*.f32 99.4%

       \[\leadsto \mathsf{fma.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), s, \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, -16\right)\right)\right)\right), s\right)\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right) \cdot s\right)} \]

7. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right) \cdot \left(-s\right)\right) \]
8. Add Preprocessing

Alternative 2: 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 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 99.4%

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

3. Simplified 99.4%

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

Alternative 3: 94.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ u \cdot \frac{1}{\frac{0.25}{s} + u \cdot \left(u \cdot \left(\frac{u \cdot -0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right) + \frac{-0.5}{s}\right)} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  u
  (/
   1.0
   (+
    (/ 0.25 s)
    (*
     u
     (+
      (* u (+ (/ (* u -0.6666666666666666) s) (/ -0.3333333333333333 s)))
      (/ -0.5 s)))))))

C

float code(float s, float u) {
	return u * (1.0f / ((0.25f / s) + (u * ((u * (((u * -0.6666666666666666f) / s) + (-0.3333333333333333f / s))) + (-0.5f / s)))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (1.0e0 / ((0.25e0 / s) + (u * ((u * (((u * (-0.6666666666666666e0)) / s) + ((-0.3333333333333333e0) / s))) + ((-0.5e0) / s)))))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(1.0) / Float32(Float32(Float32(0.25) / s) + Float32(u * Float32(Float32(u * Float32(Float32(Float32(u * Float32(-0.6666666666666666)) / s) + Float32(Float32(-0.3333333333333333) / s))) + Float32(Float32(-0.5) / s))))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (single(1.0) / ((single(0.25) / s) + (u * ((u * (((u * single(-0.6666666666666666)) / s) + (single(-0.3333333333333333) / s))) + (single(-0.5) / s)))));
end

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.1%

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

   1. flip-+ N/A

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

   2. fmm-def N/A

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

   3. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(s, 4 + u \cdot 8, \mathsf{neg}\left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)\right)}{\left(s \cdot \left(4 + u \cdot 8\right)\right) \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) - \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\mathsf{fma}\left(s, 4 + u \cdot 8, \mathsf{neg}\left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)\right)}{\left(s \cdot \left(4 + u \cdot 8\right)\right) \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) - \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto u \cdot \color{blue}{\frac{1}{\frac{s}{s \cdot \left(\left(4 + u \cdot 8\right) + \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)\right)} \cdot \frac{\left(4 + u \cdot 8\right) - \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)}{s \cdot \left(\left(4 + u \cdot 8\right) - \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)\right)}}} \]

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \color{blue}{\left(u \cdot \left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) - \frac{1}{2} \cdot \frac{1}{s}\right) + \frac{1}{4} \cdot \frac{1}{s}\right)}\right)\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \left(\frac{1}{4} \cdot \frac{1}{s} + \color{blue}{u \cdot \left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) - \frac{1}{2} \cdot \frac{1}{s}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{1}{4} \cdot \frac{1}{s}\right), \color{blue}{\left(u \cdot \left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) - \frac{1}{2} \cdot \frac{1}{s}\right)\right)}\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\frac{1}{4} \cdot 1}{s}\right), \left(\color{blue}{u} \cdot \left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) - \frac{1}{2} \cdot \frac{1}{s}\right)\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{s}\right), \left(u \cdot \left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) - \frac{1}{2} \cdot \frac{1}{s}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, s\right), \left(\color{blue}{u} \cdot \left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) - \frac{1}{2} \cdot \frac{1}{s}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, s\right), \mathsf{*.f32}\left(u, \color{blue}{\left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) - \frac{1}{2} \cdot \frac{1}{s}\right)}\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, s\right), \mathsf{*.f32}\left(u, \left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{s}\right)\right)}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, s\right), \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\left(u \cdot \left(\frac{-2}{3} \cdot \frac{u}{s} - \frac{1}{3} \cdot \frac{1}{s}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{s}\right)\right)}\right)\right)\right)\right)\right) \]

10. Simplified 95.0%

    \[\leadsto u \cdot \frac{1}{\color{blue}{\frac{0.25}{s} + u \cdot \left(u \cdot \left(\frac{u \cdot -0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right) + \frac{-0.5}{s}\right)}} \]
11. Add Preprocessing

Alternative 4: 93.4% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.1%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-+r+ N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. *-lowering-*.f32 94.3%

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

7. Applied egg-rr 94.3%

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

8. Final simplification 94.3%

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

Alternative 5: 93.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return u * (s * (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 = u * (s * (4.0e0 + (u * (8.0e0 + (u * (21.333333333333332e0 + (u * 64.0e0)))))))
end function

Julia

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

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.1%

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

6. Taylor expanded in s around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 94.2%

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

8. Simplified 94.2%

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

Alternative 6: 93.2% 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 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.1%

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

5. Simplified 94.1%

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

Alternative 7: 91.4% accurate, 7.3× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return u * ((4.0f * s) + (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 * ((4.0e0 * s) + (u * (s * (8.0e0 + (u * 21.333333333333332e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. *-lowering-*.f32 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-out N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 92.6%

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

5. Simplified 92.6%

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

6. Final simplification 92.6%

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

Alternative 8: 91.2% accurate, 8.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return u * (s * (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 = u * (s * (4.0e0 + (u * (8.0e0 + (u * 21.333333333333332e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. *-lowering-*.f32 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-out N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 92.6%

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

5. Simplified 92.6%

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

6. Taylor expanded in s around 0

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 92.5%

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

8. Simplified 92.5%

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

Alternative 9: 91.2% 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 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 92.4%

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

5. Simplified 92.4%

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

Alternative 10: 89.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ u \cdot \frac{s}{0.25 + u \cdot -0.5} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (/ s (+ 0.25 (* u -0.5)))))

C

float code(float s, float u) {
	return u * (s / (0.25f + (u * -0.5f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (s / (0.25e0 + (u * (-0.5e0))))
end function

Julia

function code(s, u)
	return Float32(u * Float32(s / Float32(Float32(0.25) + Float32(u * Float32(-0.5)))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (s / (single(0.25) + (u * single(-0.5))));
end

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.1%

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

   1. flip-+ N/A

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

   2. fmm-def N/A

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

   3. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(s, 4 + u \cdot 8, \mathsf{neg}\left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)\right)}{\left(s \cdot \left(4 + u \cdot 8\right)\right) \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) - \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\mathsf{fma}\left(s, 4 + u \cdot 8, \mathsf{neg}\left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)\right)}{\left(s \cdot \left(4 + u \cdot 8\right)\right) \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) - \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto u \cdot \color{blue}{\frac{1}{\frac{s}{s \cdot \left(\left(4 + u \cdot 8\right) + \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)\right)} \cdot \frac{\left(4 + u \cdot 8\right) - \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)}{s \cdot \left(\left(4 + u \cdot 8\right) - \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)\right)}}} \]

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{u}{s} + \frac{1}{4} \cdot \frac{1}{s}\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot \frac{u}{s}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{s}\right)}\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\frac{-1}{2} \cdot u}{s}\right), \left(\color{blue}{\frac{1}{4}} \cdot \frac{1}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot u\right), s\right), \left(\color{blue}{\frac{1}{4}} \cdot \frac{1}{s}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(u \cdot \frac{-1}{2}\right), s\right), \left(\frac{1}{4} \cdot \frac{1}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \left(\frac{1}{4} \cdot \frac{1}{s}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \left(\frac{\frac{1}{4} \cdot 1}{\color{blue}{s}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \left(\frac{\frac{1}{4}}{s}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 90.1%

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right)\right)\right)\right) \]

10. Simplified 90.1%

    \[\leadsto u \cdot \frac{1}{\color{blue}{\frac{u \cdot -0.5}{s} + \frac{0.25}{s}}} \]

11. Taylor expanded in s around 0

    \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{\left(\frac{s}{\frac{1}{4} + \frac{-1}{2} \cdot u}\right)}\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(s, \color{blue}{\left(\frac{1}{4} + \frac{-1}{2} \cdot u\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(s, \mathsf{+.f32}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{2} \cdot u\right)}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(s, \mathsf{+.f32}\left(\frac{1}{4}, \left(u \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

    4. *-lowering-*.f32 90.4%

       \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(s, \mathsf{+.f32}\left(\frac{1}{4}, \mathsf{*.f32}\left(u, \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

13. Simplified 90.4%

    \[\leadsto u \cdot \color{blue}{\frac{s}{0.25 + u \cdot -0.5}} \]
14. Add Preprocessing

Alternative 11: 87.0% accurate, 12.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return u * (s * (4.0f + (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 + (u * 8.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 88.7%

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

5. Simplified 88.7%

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

Alternative 12: 87.0% 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 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 88.7%

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

5. Simplified 88.7%

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

Alternative 13: 74.3% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 76.7%

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

5. Simplified 76.7%

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

6. Final simplification 76.7%

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

Alternative 14: 8.1% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ s \cdot -2 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s -2.0))

C

float code(float s, float u) {
	return s * -2.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (-2.0e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(-2.0))
end

MATLAB

function tmp = code(s, u)
	tmp = s * single(-2.0);
end

Derivation

1. Initial program 58.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.1%

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

   1. flip-+ N/A

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

   2. fmm-def N/A

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

   3. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(u, \left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(s, 4 + u \cdot 8, \mathsf{neg}\left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)\right)}{\left(s \cdot \left(4 + u \cdot 8\right)\right) \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) - \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\mathsf{fma}\left(s, 4 + u \cdot 8, \mathsf{neg}\left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)\right)}{\left(s \cdot \left(4 + u \cdot 8\right)\right) \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) - \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot \left(\left(s \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(u \cdot u\right)\right)}\right)}\right)\right) \]

7. Applied egg-rr 93.7%

   \[\leadsto u \cdot \color{blue}{\frac{1}{\frac{s}{s \cdot \left(\left(4 + u \cdot 8\right) + \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)\right)} \cdot \frac{\left(4 + u \cdot 8\right) - \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)}{s \cdot \left(\left(4 + u \cdot 8\right) - \left(u \cdot u\right) \cdot \left(21.333333333333332 + u \cdot 64\right)\right)}}} \]

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{u}{s} + \frac{1}{4} \cdot \frac{1}{s}\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{-1}{2} \cdot \frac{u}{s}\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{s}\right)}\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{\frac{-1}{2} \cdot u}{s}\right), \left(\color{blue}{\frac{1}{4}} \cdot \frac{1}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot u\right), s\right), \left(\color{blue}{\frac{1}{4}} \cdot \frac{1}{s}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(u \cdot \frac{-1}{2}\right), s\right), \left(\frac{1}{4} \cdot \frac{1}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \left(\frac{1}{4} \cdot \frac{1}{s}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \left(\frac{\frac{1}{4} \cdot 1}{\color{blue}{s}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \left(\frac{\frac{1}{4}}{s}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 90.1%

      \[\leadsto \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u, \frac{-1}{2}\right), s\right), \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right)\right)\right)\right) \]

10. Simplified 90.1%

    \[\leadsto u \cdot \frac{1}{\color{blue}{\frac{u \cdot -0.5}{s} + \frac{0.25}{s}}} \]

11. Taylor expanded in u around inf

    \[\leadsto \color{blue}{-2 \cdot s} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto s \cdot \color{blue}{-2} \]

    2. *-lowering-*.f32 8.2%

       \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{-2}\right) \]

13. Simplified 8.2%

    \[\leadsto \color{blue}{s \cdot -2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))))