Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.8% → 99.4%

Time: 11.3s

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

Alternative 2: 94.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	float t_0 = 8.0f + (u * (21.333333333333332f + (u * 64.0f)));
	return u * (((s * 16.0f) - (s * (t_0 * (u * (u * (8.0f + (u * 21.333333333333332f))))))) / (4.0f - (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 = 8.0e0 + (u * (21.333333333333332e0 + (u * 64.0e0)))
    code = u * (((s * 16.0e0) - (s * (t_0 * (u * (u * (8.0e0 + (u * 21.333333333333332e0))))))) / (4.0e0 - (u * t_0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-out N/A

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

7. Simplified 93.7%

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

9. Applied egg-rr 70.0%

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

10. Taylor expanded in u around 0

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

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

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

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

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

    3. *-commutative N/A

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

    4. *-lowering-*.f32 70.6%

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

12. Simplified 70.6%

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

    1. associate-/r* N/A

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

    2. associate-/r* N/A

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

    3. sub-div N/A

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

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

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

14. Applied egg-rr 94.6%

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

Alternative 3: 93.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	float t_0 = 8.0f + (u * (21.333333333333332f + (u * 64.0f)));
	return u * (s * ((16.0f - (t_0 * (u * (u * (8.0f + (u * 21.333333333333332f)))))) / (4.0f - (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 = 8.0e0 + (u * (21.333333333333332e0 + (u * 64.0e0)))
    code = u * (s * ((16.0e0 - (t_0 * (u * (u * (8.0e0 + (u * 21.333333333333332e0)))))) / (4.0e0 - (u * t_0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-out N/A

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

7. Simplified 93.7%

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

9. Applied egg-rr 70.0%

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

10. Taylor expanded in u around 0

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

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

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

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

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

    3. *-commutative N/A

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

    4. *-lowering-*.f32 70.6%

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

12. Simplified 70.6%

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

    1. *-commutative N/A

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

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

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

14. Applied egg-rr 94.3%

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

15. Final simplification 94.3%

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

Alternative 4: 93.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-out N/A

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

7. Simplified 93.7%

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

8. Final simplification 93.7%

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

Alternative 5: 93.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-out N/A

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

7. Simplified 93.7%

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

   1. +-commutative N/A

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

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

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-+l+ N/A

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

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

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

   8. distribute-lft-out N/A

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

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

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

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

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

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

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

   12. *-lowering-*.f32 93.7%

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

9. Applied egg-rr 93.7%

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

10. Final simplification 93.7%

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

Alternative 6: 93.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

   1. flip-+ N/A

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

   2. log-div N/A

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

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

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. log1p-define N/A

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

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

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

   8. swap-sqr N/A

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

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

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

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

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. log1p-define N/A

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

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

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

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

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

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

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

   19. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

8. Applied egg-rr 99.2%

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

9. Taylor expanded in s around 0

   \[\leadsto \color{blue}{s \cdot \left(\log \left(1 + 4 \cdot u\right) + -1 \cdot \log \left(1 + -16 \cdot {u}^{2}\right)\right)} \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(\log \left(1 + 4 \cdot u\right) + -1 \cdot \log \left(1 + -16 \cdot {u}^{2}\right)\right)}\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(s, \left(\log \left(1 + 4 \cdot u\right) + \left(\mathsf{neg}\left(\log \left(1 + -16 \cdot {u}^{2}\right)\right)\right)\right)\right) \]

    3. unsub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(s, \left(\log \left(1 + 4 \cdot u\right) - \color{blue}{\log \left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\log \left(1 + 4 \cdot u\right), \color{blue}{\log \left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

    5. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\left(\mathsf{log1p}\left(4 \cdot u\right)\right), \log \color{blue}{\left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\left(4 \cdot u\right)\right), \log \color{blue}{\left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \log \left(\color{blue}{1} + -16 \cdot {u}^{2}\right)\right)\right) \]

    8. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \left(\mathsf{log1p}\left(-16 \cdot {u}^{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \mathsf{log1p.f32}\left(\left(-16 \cdot {u}^{2}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \mathsf{log1p.f32}\left(\mathsf{*.f32}\left(-16, \left({u}^{2}\right)\right)\right)\right)\right) \]

    11. unpow2 N/A

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

    12. *-lowering-*.f32 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

    7. *-commutative N/A

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

    8. *-lowering-*.f32 93.5%

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

14. Simplified 93.5%

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

Alternative 7: 91.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(0 - u \cdot -4\right) - \left(u \cdot u\right) \cdot \left(-8 + u \cdot -21.333333333333332\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (- (- 0.0 (* u -4.0)) (* (* u u) (+ -8.0 (* u -21.333333333333332))))))

C

float code(float s, float u) {
	return s * ((0.0f - (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 * ((0.0e0 - (u * (-4.0e0))) - ((u * u) * ((-8.0e0) + (u * (-21.333333333333332e0)))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(0.0) - 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 * ((single(0.0) - (u * single(-4.0))) - ((u * u) * (single(-8.0) + (u * single(-21.333333333333332)))));
end

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 91.5%

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

7. Simplified 91.5%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 91.7%

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

9. Applied egg-rr 91.7%

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

10. Final simplification 91.7%

    \[\leadsto s \cdot \left(\left(0 - u \cdot -4\right) - \left(u \cdot u\right) \cdot \left(-8 + u \cdot -21.333333333333332\right)\right) \]
11. Add Preprocessing

Alternative 8: 91.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-out N/A

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

7. Simplified 93.7%

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

   1. +-commutative N/A

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

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

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

   3. associate-*l* N/A

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

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

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

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

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

   6. associate-+l+ N/A

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

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

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

   8. distribute-lft-out N/A

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

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

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

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

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

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

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

   12. *-lowering-*.f32 93.7%

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

9. Applied egg-rr 93.7%

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

10. Taylor expanded in u around 0

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

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

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

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

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

    3. *-commutative N/A

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

    4. *-lowering-*.f32 91.7%

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

12. Simplified 91.7%

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

13. Final simplification 91.7%

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

Alternative 9: 91.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. 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. *-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

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

   7. associate-*r* N/A

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

   8. distribute-rgt-out N/A

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

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

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

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

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

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

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

   12. *-lowering-*.f32 91.7%

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

7. Simplified 91.7%

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

8. Final simplification 91.7%

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

Alternative 10: 90.9% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

   1. flip-+ N/A

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

   2. log-div N/A

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

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

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. log1p-define N/A

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

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

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

   8. swap-sqr N/A

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

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

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

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

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. log1p-define N/A

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

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

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

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

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

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

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

   19. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

8. Applied egg-rr 99.2%

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

9. Taylor expanded in s around 0

   \[\leadsto \color{blue}{s \cdot \left(\log \left(1 + 4 \cdot u\right) + -1 \cdot \log \left(1 + -16 \cdot {u}^{2}\right)\right)} \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(\log \left(1 + 4 \cdot u\right) + -1 \cdot \log \left(1 + -16 \cdot {u}^{2}\right)\right)}\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(s, \left(\log \left(1 + 4 \cdot u\right) + \left(\mathsf{neg}\left(\log \left(1 + -16 \cdot {u}^{2}\right)\right)\right)\right)\right) \]

    3. unsub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(s, \left(\log \left(1 + 4 \cdot u\right) - \color{blue}{\log \left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\log \left(1 + 4 \cdot u\right), \color{blue}{\log \left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

    5. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\left(\mathsf{log1p}\left(4 \cdot u\right)\right), \log \color{blue}{\left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\left(4 \cdot u\right)\right), \log \color{blue}{\left(1 + -16 \cdot {u}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \log \left(\color{blue}{1} + -16 \cdot {u}^{2}\right)\right)\right) \]

    8. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \left(\mathsf{log1p}\left(-16 \cdot {u}^{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \mathsf{log1p.f32}\left(\left(-16 \cdot {u}^{2}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(4, u\right)\right), \mathsf{log1p.f32}\left(\mathsf{*.f32}\left(-16, \left({u}^{2}\right)\right)\right)\right)\right) \]

    11. unpow2 N/A

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

    12. *-lowering-*.f32 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in u around 0

    \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)\right)}\right) \]
13. 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 91.5%

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

14. Simplified 91.5%

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

15. Final simplification 91.5%

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

Alternative 11: 86.8% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

      \[\leadsto \left(s \cdot u\right) \cdot 4 + \left(\left(s \cdot u\right) \cdot 8\right) \cdot u \]

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 87.7%

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

7. Simplified 87.7%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f32 87.8%

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

9. Applied egg-rr 87.8%

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

10. Final simplification 87.8%

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

Alternative 12: 86.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

      \[\leadsto \left(s \cdot u\right) \cdot 4 + \left(\left(s \cdot u\right) \cdot 8\right) \cdot u \]

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f32 87.7%

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

7. Simplified 87.7%

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

Alternative 13: 73.9% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

   2. *-lowering-*.f32 73.6%

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

7. Simplified 73.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

      \[\leadsto \left(u \cdot 4\right) \cdot s \]

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

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

   5. *-lowering-*.f32 73.7%

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

9. Applied egg-rr 73.7%

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

10. Final simplification 73.7%

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

Alternative 14: 73.6% 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 61.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u around 0

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

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

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

   2. *-lowering-*.f32 73.6%

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

7. Simplified 73.6%

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

8. Final simplification 73.6%

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

Reproduce

herbie shell --seed 2024150 
(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))))))