Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 99.3%

Time: 12.5s

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: 61.1% 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: 99.3% 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-*.f32 N/A

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

   11. lower-neg.f32 99.4

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

5. Applied rewrites 99.4%

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

Alternative 2: 94.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)\\ u \cdot \frac{s \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(u, 21.333333333333332, 8\right) \cdot \left(u \cdot u\right), -16\right)}{\mathsf{fma}\left(u, t\_0, -4\right)} \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (fma u (fma u 64.0 21.333333333333332) 8.0)))
   (*
    u
    (/
     (* s (fma t_0 (* (fma u 21.333333333333332 8.0) (* u u)) -16.0))
     (fma u 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 u * ((s * fmaf(t_0, (fmaf(u, 21.333333333333332f, 8.0f) * (u * u)), -16.0f)) / fmaf(u, 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(u * Float32(Float32(s * fma(t_0, Float32(fma(u, Float32(21.333333333333332), Float32(8.0)) * Float32(u * u)), Float32(-16.0))) / fma(u, t_0, Float32(-4.0))))
end

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 93.6

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

7. Applied rewrites 93.6%

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

9. Applied rewrites 93.8%

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

10. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 94.3

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

12. Applied rewrites 94.3%

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

13. Final simplification 94.3%

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

Alternative 3: 94.5% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-fma.f32 N/A

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

   2. flip-+ N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. lower-fma.f32 N/A

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

   16. metadata-eval 93.5

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

7. Applied rewrites 93.5%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 94.2%

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

Alternative 4: 93.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 93.8

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

7. Applied rewrites 93.8%

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

Alternative 5: 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 93.6

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

7. Applied rewrites 93.6%

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

8. Final simplification 93.6%

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

Alternative 6: 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

Alternative 7: 91.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-*.f32 N/A

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

   11. lower-neg.f32 99.4

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

5. Applied rewrites 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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f32 92.1

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

8. Applied rewrites 92.1%

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

9. Final simplification 92.1%

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

Alternative 8: 91.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 93.6

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

7. Applied rewrites 93.6%

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

8. Taylor expanded in u around 0

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

   1. lower-fma.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 92.1

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

10. Applied rewrites 92.1%

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

    1. *-commutative N/A

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

    2. lift-fma.f32 N/A

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

    3. associate-*l* N/A

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

    4. distribute-lft-out N/A

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

    5. lower-*.f32 N/A

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

    6. lower-+.f32 N/A

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

    7. lower-*.f32 91.9

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

12. Applied rewrites 91.9%

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

13. Final simplification 91.9%

    \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \mathsf{fma}\left(u, 21.333333333333332, 8\right)\right)\right) \]
14. 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 93.6

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

7. Applied rewrites 93.6%

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

8. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 91.9

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

10. Applied rewrites 91.9%

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

11. Final simplification 91.9%

    \[\leadsto u \cdot \left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 21.333333333333332, 8\right), 4\right)\right) \]
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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-fma.f32 91.9

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

5. Applied rewrites 91.9%

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

Alternative 11: 86.9% accurate, 5.7× speedup

Math

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

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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-*.f32 88.1

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

5. Applied rewrites 88.1%

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

   1. lift-fma.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 88.4

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

7. Applied rewrites 88.4%

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lower-fma.f32 88.6

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

9. Applied rewrites 88.6%

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

10. Final simplification 88.6%

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

Alternative 12: 86.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-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. lower-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. lower-fma.f32 93.5

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

5. Applied rewrites 93.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 93.6

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

7. Applied rewrites 93.6%

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

8. Taylor expanded in u around 0

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

   1. lower-fma.f32 N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 92.1

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

10. Applied rewrites 92.1%

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

11. Taylor expanded in u around 0

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

    1. *-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 88.6

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

13. Applied rewrites 88.6%

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

14. Final simplification 88.6%

    \[\leadsto u \cdot \mathsf{fma}\left(s, 4, u \cdot \left(s \cdot 8\right)\right) \]
15. 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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-fma.f32 88.4

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

5. Applied rewrites 88.4%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(u, 8, 4\right)\right)} \]
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 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.f32 76.4

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

5. Applied rewrites 76.4%

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

6. Final simplification 76.4%

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

Alternative 15: 73.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.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. lower-*.f32 76.1

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

5. Applied rewrites 76.1%

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

Reproduce

herbie shell --seed 2024216 
(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))))))