Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.8% → 99.4%

Time: 10.6s

Alternatives: 12

Speedup: 21.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 60.8% 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 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

Alternative 2: 93.5% 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 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. distribute-rgt-out N/A

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

7. Simplified 93.6%

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

8. Final simplification 93.6%

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

Alternative 3: 93.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * 4.0e0) + ((u * s) * (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(Float32(u * s) * Float32(Float32(8.0) + Float32(u * Float32(Float32(21.333333333333332) + Float32(u * Float32(64.0))))))))
end

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. distribute-rgt-out N/A

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

7. Simplified 93.6%

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + \left(s \cdot u\right) \cdot \left(\left(8 + u \cdot 21.333333333333332\right) + u \cdot \left(u \cdot 64\right)\right)\right)} \]
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. *-lowering-*.f32 N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-+l+ N/A

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

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

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

   8. distribute-lft-out N/A

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

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

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

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

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

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

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

   12. *-lowering-*.f32 93.6%

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

9. Applied egg-rr 93.6%

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

10. Final simplification 93.6%

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

Alternative 4: 93.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. distribute-rgt-out N/A

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

7. Simplified 93.6%

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

   1. *-commutative N/A

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

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

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

9. Applied egg-rr 93.5%

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

10. Final simplification 93.5%

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

Alternative 5: 91.4% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

   1. remove-double-div N/A

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

   2. remove-double-neg N/A

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

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

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

   4. distribute-lft-neg-in N/A

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

   5. associate-/r* N/A

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

   6. clear-num N/A

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

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

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-*l/ N/A

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

   11. associate-/r/ N/A

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

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

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

   13. metadata-eval N/A

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

   14. frac-2neg N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in u around 0

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

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

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

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

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

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

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-in N/A

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 91.8%

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

9. Simplified 91.8%

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

10. Final simplification 91.8%

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

Alternative 6: 91.1% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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. distribute-lft-in N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-out N/A

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

   14. *-commutative N/A

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

7. Simplified 91.1%

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

   1. associate-+l+ N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-in N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

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

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

9. Applied egg-rr 91.6%

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

10. Final simplification 91.6%

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

Alternative 7: 87.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. distribute-rgt-out N/A

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

7. Simplified 93.6%

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

8. Taylor expanded in u around 0

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

10. Simplified 88.0%

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

11. Final simplification 88.0%

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

Alternative 8: 87.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(u \cdot 8\right) - \frac{u}{-0.25}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (- (* u (* u 8.0)) (/ u -0.25))))

C

float code(float s, float u) {
	return s * ((u * (u * 8.0f)) - (u / -0.25f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((u * (u * 8.0e0)) - (u / (-0.25e0)))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(u * Float32(8.0))) - Float32(u / Float32(-0.25))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * (u * single(8.0))) - (u / single(-0.25)));
end

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-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. distribute-lft-in N/A

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

   2. +-commutative N/A

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

   3. cancel-sign-sub N/A

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

   4. remove-double-div N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(s, \left(u \cdot \left(u \cdot 8\right) - \frac{1}{\frac{-1}{u}} \cdot 4\right)\right) \]

   7. associate-/r/ N/A

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

   8. clear-num N/A

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

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

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

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

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

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \left(\frac{4}{\frac{-1}{u}}\right)\right)\right) \]

   12. associate-/r/ N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \left(\frac{1}{\frac{-1}{4}} \cdot u\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \left(\frac{1}{\frac{1}{-4}} \cdot u\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \left(\frac{1}{\frac{1}{\mathsf{neg}\left(4\right)}} \cdot u\right)\right)\right) \]

   17. associate-/r/ N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \left(\frac{1}{\color{blue}{\frac{\frac{1}{\mathsf{neg}\left(4\right)}}{u}}}\right)\right)\right) \]

   18. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \left(\frac{u}{\color{blue}{\frac{1}{\mathsf{neg}\left(4\right)}}}\right)\right)\right) \]

   19. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \mathsf{/.f32}\left(u, \color{blue}{\left(\frac{1}{\mathsf{neg}\left(4\right)}\right)}\right)\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \mathsf{/.f32}\left(u, \left(\frac{1}{-4}\right)\right)\right)\right) \]

   21. metadata-eval 87.9%

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, 8\right)\right), \mathsf{/.f32}\left(u, \frac{-1}{4}\right)\right)\right) \]

9. Applied egg-rr 87.9%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(u \cdot 8\right) - \frac{u}{-0.25}\right)} \]
10. Add Preprocessing

Alternative 9: 86.9% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. *-lowering-*.f32 87.8%

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

9. Applied egg-rr 87.8%

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

10. Final simplification 87.8%

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

Alternative 10: 86.9% 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 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. +-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-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 11: 74.4% accurate, 21.8× speedup

Math

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

FPCore

(FPCore (s u) :precision binary32 (* s (/ u 0.25)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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 74.8%

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

7. Simplified 74.8%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
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. *-lowering-*.f32 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

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

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

   9. metadata-eval 75.0%

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

9. Applied egg-rr 75.0%

   \[\leadsto \color{blue}{\frac{u}{0.25} \cdot s} \]

10. Final simplification 75.0%

    \[\leadsto s \cdot \frac{u}{0.25} \]
11. Add Preprocessing

Alternative 12: 74.2% 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 59.9%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. neg-lowering-neg.f32 99.3%

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

3. Simplified 99.3%

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

5. 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 74.8%

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

7. Simplified 74.8%

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

8. Final simplification 74.8%

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

Reproduce

herbie shell --seed 2024164 
(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))))))