Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.4%

Time: 12.1s

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: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 94.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 21.333333333333332 + u \cdot 64\\ s \cdot \left(u \cdot \left(4 + \frac{\left(u \cdot u\right) \cdot \left(64 - t\_0 \cdot \left(u \cdot \left(u \cdot 21.333333333333332\right)\right)\right)}{u \cdot \left(8 - u \cdot t\_0\right)}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (+ 21.333333333333332 (* u 64.0))))
   (*
    s
    (*
     u
     (+
      4.0
      (/
       (* (* u u) (- 64.0 (* t_0 (* u (* u 21.333333333333332)))))
       (* u (- 8.0 (* u t_0)))))))))

C

float code(float s, float u) {
	float t_0 = 21.333333333333332f + (u * 64.0f);
	return s * (u * (4.0f + (((u * u) * (64.0f - (t_0 * (u * (u * 21.333333333333332f))))) / (u * (8.0f - (u * t_0))))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    t_0 = 21.333333333333332e0 + (u * 64.0e0)
    code = s * (u * (4.0e0 + (((u * u) * (64.0e0 - (t_0 * (u * (u * 21.333333333333332e0))))) / (u * (8.0e0 - (u * t_0))))))
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(21.333333333333332) + Float32(u * Float32(64.0)))
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(Float32(Float32(u * u) * Float32(Float32(64.0) - Float32(t_0 * Float32(u * Float32(u * Float32(21.333333333333332)))))) / Float32(u * Float32(Float32(8.0) - Float32(u * t_0)))))))
end

MATLAB

function tmp = code(s, u)
	t_0 = single(21.333333333333332) + (u * single(64.0));
	tmp = s * (u * (single(4.0) + (((u * u) * (single(64.0) - (t_0 * (u * (u * single(21.333333333333332)))))) / (u * (single(8.0) - (u * t_0))))));
end

Derivation

1. Initial program 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5%

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

5. Simplified 91.5%

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

   1. distribute-lft-in N/A

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

   2. flip-+ N/A

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

   3. fmm-def N/A

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

   4. *-commutative N/A

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

   5. div-sub N/A

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

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

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

7. Applied egg-rr 91.5%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 92.8%

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

10. Simplified 92.8%

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

    1. sub-div N/A

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

    2. associate-*l* N/A

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

    3. fmm-def N/A

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

    4. *-commutative N/A

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

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

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

12. Applied egg-rr 92.8%

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

Alternative 3: 93.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot \left(64 + u \cdot 170.66666666666666\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (*
   u
   (+
    4.0
    (*
     u
     (+
      8.0
      (*
       u
       (+ 21.333333333333332 (* u (+ 64.0 (* u 170.66666666666666)))))))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * (8.0f + (u * (21.333333333333332f + (u * (64.0f + (u * 170.66666666666666f)))))))));
}

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 + (u * 170.66666666666666e0)))))))))
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(Float32(64.0) + Float32(u * Float32(170.66666666666666)))))))))))
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) + (u * single(170.66666666666666))))))))));
end

Derivation

1. Initial program 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5%

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

5. Simplified 91.5%

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

   1. distribute-lft-in N/A

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

   2. flip-+ N/A

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

   3. fmm-def N/A

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

   4. *-commutative N/A

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

   5. div-sub N/A

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

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

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

7. Applied egg-rr 91.5%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 92.8%

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

10. Simplified 92.8%

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

11. Taylor expanded in u around 0

    \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot \left(64 + \frac{512}{3} \cdot u\right)\right)\right)\right)}\right)\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + u \cdot \left(64 + \frac{512}{3} \cdot u\right)\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \color{blue}{\left(u \cdot \left(\frac{64}{3} + u \cdot \left(64 + \frac{512}{3} \cdot u\right)\right)\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \color{blue}{\left(\frac{64}{3} + u \cdot \left(64 + \frac{512}{3} \cdot u\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \color{blue}{\left(u \cdot \left(64 + \frac{512}{3} \cdot u\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, \color{blue}{\left(64 + \frac{512}{3} \cdot u\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(64, \color{blue}{\left(\frac{512}{3} \cdot u\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(64, \left(u \cdot \color{blue}{\frac{512}{3}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    8. *-lowering-*.f32 92.4%

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(4, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(64, \mathsf{*.f32}\left(u, \color{blue}{\frac{512}{3}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

13. Simplified 92.4%

    \[\leadsto s \cdot \left(u \cdot \left(4 + \color{blue}{u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot \left(64 + u \cdot 170.66666666666666\right)\right)\right)}\right)\right) \]
14. Add Preprocessing

Alternative 4: 93.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5%

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

5. Simplified 91.5%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f32 91.8%

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

7. Applied egg-rr 91.8%

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

8. Final simplification 91.8%

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

Alternative 5: 93.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5%

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

5. Simplified 91.5%

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

Alternative 6: 91.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + \frac{64}{3} \cdot \left(s \cdot u\right)\right)\right)} \]
6. 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 89.6%

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

7. Simplified 89.6%

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

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

   10. *-lowering-*.f32 89.6%

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

9. Applied egg-rr 89.6%

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

10. Final simplification 89.6%

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

Alternative 7: 91.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 89.6%

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

5. Simplified 89.6%

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

6. Final simplification 89.6%

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

Alternative 8: 91.0% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in u around 0

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(8 \cdot s + \frac{64}{3} \cdot \left(s \cdot u\right)\right)\right)} \]
6. 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 89.6%

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

7. Simplified 89.6%

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

8. Taylor expanded in s around 0

   \[\leadsto \mathsf{*.f32}\left(u, \color{blue}{\left(s \cdot \left(4 + u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)\right)}\right) \]
9. 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 89.4%

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

10. Simplified 89.4%

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

11. Final simplification 89.4%

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

Alternative 9: 91.0% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 89.4%

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

5. Simplified 89.4%

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

6. Final simplification 89.4%

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

Alternative 10: 86.9% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 85.1%

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

5. Simplified 85.1%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-lowering-*.f32 85.1%

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

7. Applied egg-rr 85.1%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 85.2%

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

9. Applied egg-rr 85.2%

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

10. Final simplification 85.2%

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

Alternative 11: 86.9% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 85.1%

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

5. Simplified 85.1%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

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

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

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

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

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

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

   6. *-lowering-*.f32 85.2%

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

7. Applied egg-rr 85.2%

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

8. Final simplification 85.2%

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

Alternative 12: 86.7% 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 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 85.1%

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

5. Simplified 85.1%

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

Alternative 13: 74.0% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in u around 0

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

   1. *-lowering-*.f32 72.3%

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

5. Simplified 72.3%

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

6. Final simplification 72.3%

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

Alternative 14: 73.8% accurate, 21.8× speedup

Math

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

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(4.0) * Float32(u * s))
end

MATLAB

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

Derivation

1. Initial program 63.1%

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 72.1%

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

5. Simplified 72.1%

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

Reproduce

herbie shell --seed 2024288 
(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))))))