Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.4%

Time: 12.2s

Alternatives: 17

Speedup: 11.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. +-lft-identity N/A

      \[\leadsto \color{blue}{0 + s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)} \]

   2. mul0-rgt N/A

      \[\leadsto \color{blue}{s \cdot 0} + s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \color{blue}{s \cdot \left(0 + \log \left(\frac{1}{1 - 4 \cdot u}\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - 4 \cdot u}\right) + 0\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 - 4 \cdot u}\right) \cdot s + 0 \cdot s} \]

   6. mul0-lft N/A

      \[\leadsto \log \left(\frac{1}{1 - 4 \cdot u}\right) \cdot s + \color{blue}{0} \]

   7. log-rec N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(1 - 4 \cdot u\right)\right)\right)} \cdot s + 0 \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(1 - 4 \cdot u\right) \cdot s\right)\right)} + 0 \]

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

      \[\leadsto \color{blue}{\log \left(1 - 4 \cdot u\right) \cdot \left(\mathsf{neg}\left(s\right)\right)} + 0 \]

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - 4 \cdot u\right), \mathsf{neg}\left(s\right), 0\right)} \]

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

       \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(4\right)\right) \cdot u\right)}, \mathsf{neg}\left(s\right), 0\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\log \left(1 + \color{blue}{-4} \cdot u\right), \mathsf{neg}\left(s\right), 0\right) \]

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

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(-4 \cdot u\right)}, \mathsf{neg}\left(s\right), 0\right) \]

   14. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-4 \cdot u + 0}\right), \mathsf{neg}\left(s\right), 0\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{u \cdot -4} + 0\right), \mathsf{neg}\left(s\right), 0\right) \]

   16. accelerator-lowering-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, -4, 0\right)}\right), \mathsf{neg}\left(s\right), 0\right) \]

   17. neg-lowering-neg.f32 99.4

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -4, 0\right)\right), \color{blue}{-s}, 0\right) \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, -4, 0\right)\right), -s, 0\right)} \]

6. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. *-commutative N/A

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

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

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

   7. mul-1-neg N/A

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

   8. neg-lowering-neg.f32 99.4

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

8. Simplified 99.4%

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

Alternative 2: 94.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ s \cdot \frac{1}{\frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -0.6666666666666666, -0.3333333333333333\right), -0.5\right), 0.25\right)}{u}} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (/
   1.0
   (/
    (fma u (fma u (fma u -0.6666666666666666 -0.3333333333333333) -0.5) 0.25)
    u))))

C

float code(float s, float u) {
	return s * (1.0f / (fmaf(u, fmaf(u, fmaf(u, -0.6666666666666666f, -0.3333333333333333f), -0.5f), 0.25f) / u));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(1.0) / Float32(fma(u, fma(u, fma(u, Float32(-0.6666666666666666), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(0.25)) / u)))
end

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.1

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

5. Simplified 94.1%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

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

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

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

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

7. Applied egg-rr 93.9%

   \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right) \cdot \left(u \cdot u\right), -16\right) \cdot u}}} \]

8. Taylor expanded in u around 0

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

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

      \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{\frac{1}{4} + u \cdot \left(u \cdot \left(\frac{-2}{3} \cdot u - \frac{1}{3}\right) - \frac{1}{2}\right)}{u}}} \]

   2. +-commutative N/A

      \[\leadsto s \cdot \frac{1}{\frac{\color{blue}{u \cdot \left(u \cdot \left(\frac{-2}{3} \cdot u - \frac{1}{3}\right) - \frac{1}{2}\right) + \frac{1}{4}}}{u}} \]

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

      \[\leadsto s \cdot \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(u, u \cdot \left(\frac{-2}{3} \cdot u - \frac{1}{3}\right) - \frac{1}{2}, \frac{1}{4}\right)}}{u}} \]

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

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

      \[\leadsto s \cdot \frac{1}{\frac{\mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{-2}{3} \cdot u - \frac{1}{3}, \frac{-1}{2}\right)}, \frac{1}{4}\right)}{u}} \]

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f32 95.2

       \[\leadsto s \cdot \frac{1}{\frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, -0.6666666666666666, -0.3333333333333333\right)}, -0.5\right), 0.25\right)}{u}} \]

10. Simplified 95.2%

    \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -0.6666666666666666, -0.3333333333333333\right), -0.5\right), 0.25\right)}{u}}} \]
11. Add Preprocessing

Alternative 3: 93.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.1

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

5. Simplified 94.1%

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

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

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

   11. accelerator-lowering-fma.f32 N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 94.7

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

7. Applied egg-rr 94.7%

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

Alternative 4: 93.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.1

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

5. Simplified 94.1%

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

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

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

   11. accelerator-lowering-fma.f32 N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 94.7

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

7. Applied egg-rr 94.7%

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

8. Taylor expanded in u around 0

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

   1. distribute-lft-in N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-in N/A

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

   11. associate-+r+ N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. *-commutative N/A

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

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

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

10. Simplified 94.7%

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

Alternative 5: 93.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.1

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

5. Simplified 94.1%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 94.3

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

7. Applied egg-rr 94.3%

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

Alternative 6: 93.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

      \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right) + \color{blue}{{u}^{2}} \cdot \left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right) + \color{blue}{\left(\frac{64}{3} \cdot s + 64 \cdot \left(s \cdot u\right)\right) \cdot {u}^{2}}\right) \]

   10. *-commutative N/A

       \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right) + \left(\frac{64}{3} \cdot s + 64 \cdot \color{blue}{\left(u \cdot s\right)}\right) \cdot {u}^{2}\right) \]

   11. associate-*r* N/A

       \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right) + \left(\frac{64}{3} \cdot s + \color{blue}{\left(64 \cdot u\right) \cdot s}\right) \cdot {u}^{2}\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right) + \color{blue}{\left(s \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)} \cdot {u}^{2}\right) \]

   13. associate-*l* N/A

       \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right) + \color{blue}{s \cdot \left(\left(\frac{64}{3} + 64 \cdot u\right) \cdot {u}^{2}\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto u \cdot \left(s \cdot \left(4 + 8 \cdot u\right) + s \cdot \color{blue}{\left({u}^{2} \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)}\right) \]

   15. unpow2 N/A

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

5. Simplified 94.1%

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

Alternative 7: 93.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.1

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

5. Simplified 94.1%

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

Alternative 8: 91.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.1

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

5. Simplified 94.1%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   7. accelerator-lowering-fma.f32 94.1

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

7. Applied egg-rr 94.1%

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

8. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

   10. *-lowering-*.f32 91.9

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

10. Simplified 91.9%

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

11. Final simplification 91.9%

    \[\leadsto u \cdot \mathsf{fma}\left(u, s \cdot \mathsf{fma}\left(u, 21.333333333333332, 8\right), s \cdot 4\right) \]
12. Add Preprocessing

Alternative 9: 91.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 91.6

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

5. Simplified 91.6%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. *-lowering-*.f32 91.8

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

7. Applied egg-rr 91.8%

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

Alternative 10: 91.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 91.6

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

5. Simplified 91.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. accelerator-lowering-fma.f32 91.6

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

7. Applied egg-rr 91.6%

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

8. Final simplification 91.6%

   \[\leadsto u \cdot \left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 21.333333333333332, 8\right), 4\right)\right) \]
9. Add Preprocessing

Alternative 11: 91.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 91.6

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

5. Simplified 91.6%

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

Alternative 12: 87.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f32 86.8

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

5. Simplified 86.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. accelerator-lowering-fma.f32 86.9

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

7. Applied egg-rr 86.9%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. *-lowering-*.f32 87.0

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

9. Applied egg-rr 87.0%

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

10. Final simplification 87.0%

    \[\leadsto u \cdot \mathsf{fma}\left(u \cdot 8, s, s \cdot 4\right) \]
11. Add Preprocessing

Alternative 13: 87.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f32 86.8

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

5. Simplified 86.8%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. *-lowering-*.f32 87.0

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

7. Applied egg-rr 87.0%

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

Alternative 14: 86.9% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f32 86.9

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

5. Simplified 86.9%

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

6. Final simplification 86.9%

   \[\leadsto u \cdot \left(s \cdot \mathsf{fma}\left(u, 8, 4\right)\right) \]
7. Add Preprocessing

Alternative 15: 86.9% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f32 86.8

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

5. Simplified 86.8%

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

Alternative 16: 74.0% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in u around 0

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

   1. *-lowering-*.f32 73.7

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

5. Simplified 73.7%

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

6. Final simplification 73.7%

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

Alternative 17: 73.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.9%

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f32 73.4

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

5. Simplified 73.4%

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

Reproduce

herbie shell --seed 2024195 
(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))))))