Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 99.4%

Time: 10.9s

Alternatives: 12

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: 61.2% 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 99.3%

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

3. Simplified 99.3%

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

Alternative 2: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\\ t_1 := u \cdot t\_0\\ s \cdot \frac{u}{\frac{\left(u \cdot u\right) \cdot \left(t\_0 \cdot \left(8 + u \cdot 21.333333333333332\right)\right) + \left(16 + \left(u \cdot -4\right) \cdot t\_0\right)}{64 + t\_0 \cdot \left(t\_1 \cdot \left(u \cdot t\_1\right)\right)}} \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (+ 8.0 (* u (+ 21.333333333333332 (* u 64.0))))) (t_1 (* u t_0)))
   (*
    s
    (/
     u
     (/
      (+
       (* (* u u) (* t_0 (+ 8.0 (* u 21.333333333333332))))
       (+ 16.0 (* (* u -4.0) t_0)))
      (+ 64.0 (* t_0 (* t_1 (* u t_1)))))))))

C

float code(float s, float u) {
	float t_0 = 8.0f + (u * (21.333333333333332f + (u * 64.0f)));
	float t_1 = u * t_0;
	return s * (u / ((((u * u) * (t_0 * (8.0f + (u * 21.333333333333332f)))) + (16.0f + ((u * -4.0f) * t_0))) / (64.0f + (t_0 * (t_1 * (u * t_1))))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    real(4) :: t_1
    t_0 = 8.0e0 + (u * (21.333333333333332e0 + (u * 64.0e0)))
    t_1 = u * t_0
    code = s * (u / ((((u * u) * (t_0 * (8.0e0 + (u * 21.333333333333332e0)))) + (16.0e0 + ((u * (-4.0e0)) * t_0))) / (64.0e0 + (t_0 * (t_1 * (u * t_1))))))
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(8.0) + Float32(u * Float32(Float32(21.333333333333332) + Float32(u * Float32(64.0)))))
	t_1 = Float32(u * t_0)
	return Float32(s * Float32(u / Float32(Float32(Float32(Float32(u * u) * Float32(t_0 * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332))))) + Float32(Float32(16.0) + Float32(Float32(u * Float32(-4.0)) * t_0))) / Float32(Float32(64.0) + Float32(t_0 * Float32(t_1 * Float32(u * t_1)))))))
end

MATLAB

function tmp = code(s, u)
	t_0 = single(8.0) + (u * (single(21.333333333333332) + (u * single(64.0))));
	t_1 = u * t_0;
	tmp = s * (u / ((((u * u) * (t_0 * (single(8.0) + (u * single(21.333333333333332))))) + (single(16.0) + ((u * single(-4.0)) * t_0))) / (single(64.0) + (t_0 * (t_1 * (u * t_1))))));
end

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.5%

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

5. Simplified 94.5%

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

   1. +-commutative N/A

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

   2. flip3-+ N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

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

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

7. Applied egg-rr 94.4%

   \[\leadsto s \cdot \left(u \cdot \color{blue}{\frac{64 + \left(u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right) \cdot \left(\left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)}{\left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right) + \left(16 - \left(4 \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)}}\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(u, \mathsf{/.f32}\left(\mathsf{+.f32}\left(64, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \color{blue}{\frac{64}{3}}\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right)\right)\right), \mathsf{\_.f32}\left(16, \mathsf{*.f32}\left(\mathsf{*.f32}\left(4, u\right), \mathsf{+.f32}\left(8, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{64}{3}, \mathsf{*.f32}\left(u, 64\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 95.4%

    \[\leadsto s \cdot \left(u \cdot \frac{64 + \left(u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right) \cdot \left(\left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)}{\left(8 + u \cdot \color{blue}{21.333333333333332}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right) + \left(16 - \left(4 \cdot u\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(s, \left(u \cdot \frac{1}{\color{blue}{\frac{\left(8 + u \cdot \frac{64}{3}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) + \left(16 - \left(4 \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)}{64 + \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) \cdot \left(\left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)\right)}}}\right)\right) \]

    2. un-div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(s, \left(\frac{u}{\color{blue}{\frac{\left(8 + u \cdot \frac{64}{3}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) + \left(16 - \left(4 \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)}{64 + \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) \cdot \left(\left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)\right)}}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(u, \color{blue}{\left(\frac{\left(8 + u \cdot \frac{64}{3}\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) + \left(16 - \left(4 \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)}{64 + \left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right) \cdot \left(\left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right) \cdot \left(\left(u \cdot u\right) \cdot \left(8 + u \cdot \left(\frac{64}{3} + u \cdot 64\right)\right)\right)\right)}\right)}\right)\right) \]

12. Applied egg-rr 95.6%

    \[\leadsto s \cdot \color{blue}{\frac{u}{\frac{\left(u \cdot u\right) \cdot \left(\left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right) + \left(16 + \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(u \cdot -4\right)\right)}{64 + \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(\left(u \cdot \left(u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right) \cdot \left(u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)}}} \]

13. Final simplification 95.6%

    \[\leadsto s \cdot \frac{u}{\frac{\left(u \cdot u\right) \cdot \left(\left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right) + \left(16 + \left(u \cdot -4\right) \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)}{64 + \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right) \cdot \left(\left(u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right) \cdot \left(u \cdot \left(u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)\right)}} \]
14. Add Preprocessing

Alternative 3: 94.4% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (+ 21.333333333333332 (* u 64.0))))
   (*
    s
    (*
     u
     (+
      4.0
      (/
       (* u (- 64.0 (* t_0 (* u (* u 21.333333333333332)))))
       (- 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 * (64.0f - (t_0 * (u * (u * 21.333333333333332f))))) / (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 * (64.0e0 - (t_0 * (u * (u * 21.333333333333332e0))))) / (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(u * Float32(Float32(64.0) - Float32(t_0 * Float32(u * Float32(u * Float32(21.333333333333332)))))) / 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 * (single(64.0) - (t_0 * (u * (u * single(21.333333333333332)))))) / (single(8.0) - (u * t_0)))));
end

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.5%

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

5. Simplified 94.5%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

7. Applied egg-rr 94.5%

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

8. Taylor expanded in u around 0

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 4: 93.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.5%

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

5. Simplified 94.5%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)} \]
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-lft-in N/A

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

   3. *-commutative N/A

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

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

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

   5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f32 94.7%

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

7. Applied egg-rr 94.7%

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

8. Final simplification 94.7%

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

Alternative 5: 93.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.5%

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

5. Simplified 94.5%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f32 94.5%

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

7. Applied egg-rr 94.5%

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

Alternative 6: 93.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.5%

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

5. Simplified 94.5%

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

6. Final simplification 94.5%

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

Alternative 7: 91.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.5%

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

5. Simplified 94.5%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)} \]
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-lft-in N/A

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

   3. *-commutative N/A

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

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

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

   5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f32 94.7%

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

7. Applied egg-rr 94.7%

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

8. Taylor expanded in u around 0

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

10. Simplified 92.9%

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

11. Final simplification 92.9%

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

Alternative 8: 91.1% 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 92.8%

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

5. Simplified 92.8%

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

Alternative 9: 87.0% accurate, 9.9× speedup

Math

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

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(Float32(u * 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.5%

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

5. Simplified 94.5%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)} \]
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-lft-in N/A

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

   3. *-commutative N/A

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

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

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

   5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f32 94.7%

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

7. Applied egg-rr 94.7%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 89.0%

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

10. Simplified 89.0%

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

11. Final simplification 89.0%

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

Alternative 10: 86.8% 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 88.8%

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

5. Simplified 88.8%

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

Alternative 11: 74.2% 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 76.5%

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

5. Simplified 76.5%

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

6. Final simplification 76.5%

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

Alternative 12: 74.0% 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 76.3%

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

5. Simplified 76.3%

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

Reproduce

herbie shell --seed 2024161 
(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))))))