Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.4%

Time: 13.0s

Alternatives: 15

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 59.9%

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

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

   2. log-rec N/A

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

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

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. neg-lowering-neg.f32 99.4

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

5. Simplified 99.4%

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

Alternative 2: 94.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, 341.3333333333333, u \cdot \left(s \cdot 967.1111111111111\right)\right), s \cdot 64\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (/
  (*
   u
   (fma
    s
    -16.0
    (*
     (* u u)
     (fma
      u
      (fma s 341.3333333333333 (* u (* s 967.1111111111111)))
      (* s 64.0)))))
  (fma u (fma u (fma u 64.0 21.333333333333332) 8.0) -4.0)))

C

float code(float s, float u) {
	return (u * fmaf(s, -16.0f, ((u * u) * fmaf(u, fmaf(s, 341.3333333333333f, (u * (s * 967.1111111111111f))), (s * 64.0f))))) / fmaf(u, fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f), -4.0f);
}

Julia

function code(s, u)
	return Float32(Float32(u * fma(s, Float32(-16.0), Float32(Float32(u * u) * fma(u, fma(s, Float32(341.3333333333333), Float32(u * Float32(s * Float32(967.1111111111111)))), Float32(s * Float32(64.0)))))) / fma(u, fma(u, fma(u, Float32(64.0), Float32(21.333333333333332)), Float32(8.0)), Float32(-4.0)))
end

Derivation

1. Initial program 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.2

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

5. Simplified 94.2%

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

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

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

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

7. Applied egg-rr 94.1%

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

8. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f32 94.6

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

10. Simplified 94.6%

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

11. Taylor expanded in u around 0

    \[\leadsto \frac{\color{blue}{u \cdot \left(-16 \cdot s + {u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]
12. Step-by-step derivation

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

       \[\leadsto \frac{\color{blue}{u \cdot \left(-16 \cdot s + {u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{u \cdot \left(\color{blue}{s \cdot -16} + {u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

       \[\leadsto \frac{u \cdot \color{blue}{\mathsf{fma}\left(s, -16, {u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

       \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \color{blue}{{u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)}\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \color{blue}{\left(u \cdot u\right)} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

       \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \color{blue}{\left(u \cdot u\right)} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    7. +-commutative N/A

       \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \color{blue}{\left(u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right) + 64 \cdot s\right)}\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

       \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \color{blue}{\mathsf{fma}\left(u, \frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right), 64 \cdot s\right)}\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    9. *-commutative N/A

       \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \color{blue}{s \cdot \frac{1024}{3}} + \frac{8704}{9} \cdot \left(s \cdot u\right), 64 \cdot s\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(s, \frac{1024}{3}, \frac{8704}{9} \cdot \left(s \cdot u\right)\right)}, 64 \cdot s\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    11. associate-*r* N/A

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, \frac{1024}{3}, \color{blue}{\left(\frac{8704}{9} \cdot s\right) \cdot u}\right), 64 \cdot s\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    12. *-commutative N/A

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, \frac{1024}{3}, \color{blue}{u \cdot \left(\frac{8704}{9} \cdot s\right)}\right), 64 \cdot s\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, \frac{1024}{3}, \color{blue}{u \cdot \left(\frac{8704}{9} \cdot s\right)}\right), 64 \cdot s\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    14. *-commutative N/A

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, \frac{1024}{3}, u \cdot \color{blue}{\left(s \cdot \frac{8704}{9}\right)}\right), 64 \cdot s\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, \frac{1024}{3}, u \cdot \color{blue}{\left(s \cdot \frac{8704}{9}\right)}\right), 64 \cdot s\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    16. *-commutative N/A

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, \frac{1024}{3}, u \cdot \left(s \cdot \frac{8704}{9}\right)\right), \color{blue}{s \cdot 64}\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    17. *-lowering-*.f32 95.0

        \[\leadsto \frac{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, 341.3333333333333, u \cdot \left(s \cdot 967.1111111111111\right)\right), \color{blue}{s \cdot 64}\right)\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \]

13. Simplified 95.0%

    \[\leadsto \frac{\color{blue}{u \cdot \mathsf{fma}\left(s, -16, \left(u \cdot u\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s, 341.3333333333333, u \cdot \left(s \cdot 967.1111111111111\right)\right), s \cdot 64\right)\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \]
14. Add Preprocessing

Alternative 3: 93.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 967.1111111111111, 341.3333333333333\right), 64\right), -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (/
  (*
   (* u s)
   (fma
    (* u u)
    (fma u (fma u 967.1111111111111 341.3333333333333) 64.0)
    -16.0))
  (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 * u), fmaf(u, fmaf(u, 967.1111111111111f, 341.3333333333333f), 64.0f), -16.0f)) / fmaf(u, fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f), -4.0f);
}

Julia

function code(s, u)
	return Float32(Float32(Float32(u * s) * fma(Float32(u * u), fma(u, fma(u, Float32(967.1111111111111), Float32(341.3333333333333)), Float32(64.0)), Float32(-16.0))) / fma(u, fma(u, fma(u, Float32(64.0), Float32(21.333333333333332)), Float32(8.0)), Float32(-4.0)))
end

Derivation

1. Initial program 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.2

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

5. Simplified 94.2%

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

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

   3. flip-+ N/A

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

   4. associate-*r/ N/A

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

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

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

7. Applied egg-rr 94.1%

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

8. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f32 94.6

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

10. Simplified 94.6%

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

11. Taylor expanded in u around 0

    \[\leadsto \frac{\left(u \cdot s\right) \cdot \color{blue}{\left({u}^{2} \cdot \left(64 + u \cdot \left(\frac{1024}{3} + \frac{8704}{9} \cdot u\right)\right) - 16\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]
12. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \color{blue}{\left({u}^{2} \cdot \left(64 + u \cdot \left(\frac{1024}{3} + \frac{8704}{9} \cdot u\right)\right) + \left(\mathsf{neg}\left(16\right)\right)\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \left({u}^{2} \cdot \left(64 + u \cdot \left(\frac{1024}{3} + \frac{8704}{9} \cdot u\right)\right) + \color{blue}{-16}\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left({u}^{2}, 64 + u \cdot \left(\frac{1024}{3} + \frac{8704}{9} \cdot u\right), -16\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    4. unpow2 N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{u \cdot u}, 64 + u \cdot \left(\frac{1024}{3} + \frac{8704}{9} \cdot u\right), -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(\color{blue}{u \cdot u}, 64 + u \cdot \left(\frac{1024}{3} + \frac{8704}{9} \cdot u\right), -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    6. +-commutative N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(u \cdot u, \color{blue}{u \cdot \left(\frac{1024}{3} + \frac{8704}{9} \cdot u\right) + 64}, -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

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

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(u \cdot u, \color{blue}{\mathsf{fma}\left(u, \frac{1024}{3} + \frac{8704}{9} \cdot u, 64\right)}, -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    8. +-commutative N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \color{blue}{\frac{8704}{9} \cdot u + \frac{1024}{3}}, 64\right), -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    9. *-commutative N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \color{blue}{u \cdot \frac{8704}{9}} + \frac{1024}{3}, 64\right), -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    10. accelerator-lowering-fma.f32 94.8

        \[\leadsto \frac{\left(u \cdot s\right) \cdot \mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, 967.1111111111111, 341.3333333333333\right)}, 64\right), -16\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \]

13. Simplified 94.8%

    \[\leadsto \frac{\left(u \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 967.1111111111111, 341.3333333333333\right), 64\right), -16\right)}}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \]
14. Add Preprocessing

Alternative 4: 93.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.2

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

5. Simplified 94.2%

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

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

   3. distribute-rgt-in N/A

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

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

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

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

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

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

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

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

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

   13. *-lowering-*.f32 94.6

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

7. Applied egg-rr 94.6%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-out N/A

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f32 94.6

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

9. Applied egg-rr 94.6%

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

Alternative 5: 93.4% 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 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.2

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

5. Simplified 94.2%

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

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f32 94.6

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

7. Applied egg-rr 94.6%

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

Alternative 6: 93.1% 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 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.2

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

5. Simplified 94.2%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), 4\right)\right)} \]
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 94.3

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

7. Applied egg-rr 94.3%

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

8. Final simplification 94.3%

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

Alternative 7: 93.1% 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 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 94.2

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

5. Simplified 94.2%

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

Alternative 8: 91.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

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

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

   2. log-rec N/A

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

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

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. neg-lowering-neg.f32 99.4

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

5. Simplified 99.4%

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

6. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. *-lowering-*.f32 92.5

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

8. Simplified 92.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. accelerator-lowering-fma.f32 92.5

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

10. Applied egg-rr 92.5%

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

Alternative 9: 91.0% 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 59.9%

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

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

   2. log-rec N/A

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

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

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. neg-lowering-neg.f32 99.4

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

5. Simplified 99.4%

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

6. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. *-lowering-*.f32 92.5

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

8. Simplified 92.5%

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

9. Taylor expanded in u around 0

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

    1. distribute-rgt-in N/A

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

    2. associate-*r* N/A

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

    3. associate-+r+ N/A

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

    4. associate-*r* N/A

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

    5. associate-*r* N/A

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

    6. unpow2 N/A

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

    7. *-commutative N/A

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

    8. associate-+r+ N/A

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

    9. *-commutative N/A

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

    10. *-commutative N/A

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

    11. unpow2 N/A

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

    12. associate-*r* N/A

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

    13. associate-*r* N/A

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

    14. *-commutative N/A

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

    15. associate-*l* N/A

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

    16. distribute-lft-out N/A

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

    17. associate-*r* N/A

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

    18. *-commutative N/A

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

    19. distribute-rgt-in N/A

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

11. Simplified 92.3%

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

Alternative 10: 91.0% 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 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 92.2

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

5. Simplified 92.2%

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

Alternative 11: 86.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.9%

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

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

   2. log-rec N/A

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

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

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. neg-lowering-neg.f32 99.4

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

5. Simplified 99.4%

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

6. Taylor expanded in u around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out N/A

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

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

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

   13. *-lowering-*.f32 92.5

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

8. Simplified 92.5%

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

9. Taylor expanded in u around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f32 88.2

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

11. Simplified 88.2%

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

12. Final simplification 88.2%

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

Alternative 12: 86.7% 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 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-commutative N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

   13. *-lowering-*.f32 87.6

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

5. Simplified 87.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. accelerator-lowering-fma.f32 88.0

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

7. Applied egg-rr 88.0%

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

8. Final simplification 88.0%

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

Alternative 13: 86.7% 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 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 87.9

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

5. Simplified 87.9%

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

Alternative 14: 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 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 75.4

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

5. Simplified 75.4%

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

6. Final simplification 75.4%

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

Alternative 15: 73.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 75.2

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

5. Simplified 75.2%

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

6. Final simplification 75.2%

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

Reproduce

herbie shell --seed 2024199 
(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))))))