Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.7% → 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: 61.7% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 94.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(455.1111111111111 \cdot s, \frac{s}{\left(21.333333333333332 - 64 \cdot u\right) \cdot s}, \frac{\left(4096 \cdot \left(\left(s \cdot u\right) \cdot s\right)\right) \cdot u}{\left(64 \cdot u - 21.333333333333332\right) \cdot s}\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (fma
   (fma
    (fma
     (* 455.1111111111111 s)
     (/ s (* (- 21.333333333333332 (* 64.0 u)) s))
     (/
      (* (* 4096.0 (* (* s u) s)) u)
      (* (- (* 64.0 u) 21.333333333333332) s)))
    u
    (* 8.0 s))
   u
   (* 4.0 s))
  u))

C

float code(float s, float u) {
	return fmaf(fmaf(fmaf((455.1111111111111f * s), (s / ((21.333333333333332f - (64.0f * u)) * s)), (((4096.0f * ((s * u) * s)) * u) / (((64.0f * u) - 21.333333333333332f) * s))), u, (8.0f * s)), u, (4.0f * s)) * u;
}

Julia

function code(s, u)
	return Float32(fma(fma(fma(Float32(Float32(455.1111111111111) * s), Float32(s / Float32(Float32(Float32(21.333333333333332) - Float32(Float32(64.0) * u)) * s)), Float32(Float32(Float32(Float32(4096.0) * Float32(Float32(s * u) * s)) * u) / Float32(Float32(Float32(Float32(64.0) * u) - Float32(21.333333333333332)) * s))), u, Float32(Float32(8.0) * s)), u, Float32(Float32(4.0) * s)) * u)
end

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 95.1

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

7. Applied rewrites 95.1%

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

9. Applied rewrites 92.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{455.1111111111111 \cdot \left(s \cdot s\right) - \left(\left(s \cdot u\right) \cdot \left(s \cdot u\right)\right) \cdot 4096}{21.333333333333332 \cdot s - \left(s \cdot u\right) \cdot 64}, u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u \]
10. Step-by-step derivation

11. Applied rewrites 95.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(455.1111111111111 \cdot s, \frac{s}{s \cdot \left(21.333333333333332 - 64 \cdot u\right)}, -\frac{\left(4096 \cdot \left(\left(s \cdot u\right) \cdot s\right)\right) \cdot u}{s \cdot \left(21.333333333333332 - 64 \cdot u\right)}\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u \]

12. Final simplification 95.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(455.1111111111111 \cdot s, \frac{s}{\left(21.333333333333332 - 64 \cdot u\right) \cdot s}, \frac{\left(4096 \cdot \left(\left(s \cdot u\right) \cdot s\right)\right) \cdot u}{\left(64 \cdot u - 21.333333333333332\right) \cdot s}\right), u, 8 \cdot s\right), u, 4 \cdot s\right) \cdot u \]
13. Add Preprocessing

Alternative 3: 93.8% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 95.1

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

7. Applied rewrites 95.1%

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

9. Applied rewrites 94.8%

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

11. Applied rewrites 95.5%

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

12. Final simplification 95.5%

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

Alternative 4: 93.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 95.1

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

7. Applied rewrites 95.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 95.1%

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

Alternative 5: 93.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.2%

   \[\leadsto \left(-\color{blue}{\left(\mathsf{log1p}\left(-\left(16 \cdot u\right) \cdot u\right) - \mathsf{log1p}\left(4 \cdot u\right)\right)}\right) \cdot s \]

7. Taylor expanded in u around 0

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

9. Applied rewrites 94.7%

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

11. Applied rewrites 94.7%

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

Alternative 6: 93.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 94.7%

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

6. Final simplification 94.7%

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

Alternative 7: 91.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 93.1

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

5. Applied rewrites 93.1%

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

7. Applied rewrites 93.4%

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

8. Final simplification 93.4%

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

Alternative 8: 91.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.2%

   \[\leadsto \left(-\color{blue}{\left(\mathsf{log1p}\left(-\left(16 \cdot u\right) \cdot u\right) - \mathsf{log1p}\left(4 \cdot u\right)\right)}\right) \cdot s \]

7. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. lower-*.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 93.4

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

9. Applied rewrites 93.4%

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

10. Final simplification 93.4%

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

Alternative 9: 90.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. distribute-lft-out N/A

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

   10. associate-*l* N/A

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

   11. distribute-lft-in N/A

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

   12. lower-*.f32 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 93.1

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

5. Applied rewrites 93.1%

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

6. Final simplification 93.1%

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

Alternative 10: 86.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 95.1

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

7. Applied rewrites 95.1%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 89.5%

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

Alternative 11: 86.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 59.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lift--.f32 N/A

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

   9. sub-neg N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 89.5

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

7. Applied rewrites 89.5%

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

8. Final simplification 89.5%

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

Alternative 12: 86.5% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 89.2

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

5. Applied rewrites 89.2%

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

Alternative 13: 73.5% accurate, 11.4× speedup

Math

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

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(Float32(4.0) * u) * s)
end

MATLAB

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

Derivation

1. Initial program 59.2%

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

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

   2. lower-*.f32 76.5

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

5. Applied rewrites 76.5%

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

6. Final simplification 76.5%

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

Alternative 14: 73.3% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 89.2

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

5. Applied rewrites 89.2%

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

7. Applied rewrites 88.9%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 76.3%

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

Reproduce

herbie shell --seed 2024235 
(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))))))