Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.0% → 99.4%

Time: 10.3s

Alternatives: 14

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: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.1× 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 99.3

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

5. Simplified 99.3%

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

Alternative 2: 94.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	float t_0 = fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), -8.0f);
	return s * (u * fmaf(u, ((((u * fmaf(u, 64.0f, 21.333333333333332f)) * (u * 21.333333333333332f)) / t_0) - (64.0f / t_0)), 4.0f));
}

Julia

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

Derivation

1. Initial program 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5

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

5. Simplified 91.5%

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

   1. flip-+ N/A

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

   2. metadata-eval N/A

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

   3. div-sub N/A

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

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

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

7. Applied egg-rr 91.5%

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

8. Taylor expanded in u around 0

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

10. Simplified 93.0%

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

Alternative 3: 94.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5

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

5. Simplified 91.5%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

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

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

7. Applied egg-rr 91.5%

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

8. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. accelerator-lowering-fma.f32 93.0

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

10. Simplified 93.0%

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

Alternative 4: 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5

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

5. Simplified 91.5%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

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

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

7. Applied egg-rr 91.5%

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

   1. associate-/r/ N/A

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

   2. /-rgt-identity N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

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

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

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

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

   15. *-lowering-*.f32 91.7

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

9. Applied egg-rr 91.7%

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

Alternative 5: 93.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5

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

5. Simplified 91.5%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

   7. accelerator-lowering-fma.f32 91.7

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

7. Applied egg-rr 91.7%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(u, 4, u \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)\right)\right)} \]
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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5

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

5. Simplified 91.5%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), 4\right)\right)} \]
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 91.6

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

7. Applied egg-rr 91.6%

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

8. Final simplification 91.6%

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

Alternative 7: 93.2% 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 91.5

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

5. Simplified 91.5%

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

Alternative 8: 91.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 89.7

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

5. Simplified 89.7%

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

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 89.9

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

7. Applied egg-rr 89.9%

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

Alternative 9: 91.2% 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 89.7

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

5. Simplified 89.7%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 21.333333333333332, 8\right), 4\right)\right)} \]
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 89.8

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

7. Applied egg-rr 89.8%

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

8. Final simplification 89.8%

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

Alternative 10: 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 89.7

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

5. Simplified 89.7%

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

Alternative 11: 87.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 85.6

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

5. Simplified 85.6%

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 85.7

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

7. Applied egg-rr 85.7%

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

Alternative 12: 87.0% 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 85.6

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

5. Simplified 85.6%

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

Alternative 13: 74.2% 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.2

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

5. Simplified 72.2%

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

6. Final simplification 72.2%

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

Alternative 14: 74.0% 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 71.9

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

5. Simplified 71.9%

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

Reproduce

herbie shell --seed 2024198 
(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))))))