Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 62.0% → 99.4%

Time: 10.4s

Alternatives: 10

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: 62.0% 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 62.5%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. log-rec N/A

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

   3. distribute-lft-neg-out N/A

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

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

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

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

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

   6. cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. neg-lowering-neg.f32 99.4%

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

5. Simplified 99.4%

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

Alternative 2: 94.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 93.7%

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

5. Simplified 93.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

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

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

7. Applied egg-rr 93.7%

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

8. Taylor expanded in u around 0

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

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

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

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

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

   12. *-commutative N/A

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

   13. *-lowering-*.f32 94.9%

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

10. Simplified 94.9%

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

Alternative 3: 92.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 93.7%

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

5. Simplified 93.7%

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

Alternative 4: 92.5% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 93.7%

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

5. Simplified 93.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

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

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

7. Applied egg-rr 93.7%

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

8. Taylor expanded in u around 0

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

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

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 93.4%

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

10. Simplified 93.4%

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

Alternative 5: 90.7% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 91.7%

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

5. Simplified 91.7%

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

Alternative 6: 88.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 93.7%

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

5. Simplified 93.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

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

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

7. Applied egg-rr 93.7%

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

8. Taylor expanded in u around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 89.0%

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

10. Simplified 89.0%

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

Alternative 7: 86.3% 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 62.5%

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

3. Taylor expanded in u around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 86.8%

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

5. Simplified 86.8%

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

Alternative 8: 73.4% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u around 0

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

   1. *-lowering-*.f32 73.1%

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

5. Simplified 73.1%

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

6. Final simplification 73.1%

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

Alternative 9: 73.2% 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 62.5%

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

3. Taylor expanded in u around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 72.8%

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

5. Simplified 72.8%

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

Alternative 10: 8.2% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in u around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 93.7%

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

5. Simplified 93.7%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

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

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

7. Applied egg-rr 93.7%

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

8. Taylor expanded in u around 0

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 89.0%

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

10. Simplified 89.0%

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

11. Taylor expanded in u around inf

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

    1. *-commutative N/A

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

    2. *-lowering-*.f32 8.5%

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

13. Simplified 8.5%

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

Reproduce

herbie shell --seed 2024192 
(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))))))