Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.7% → 99.4%

Time: 11.8s

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: 60.7% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.1%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. log-rec N/A

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

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

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

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

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

   5. 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. Simplified 99.4%

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

Alternative 2: 94.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot \mathsf{fma}\left(u, u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u \cdot s, 967.1111111111111, s \cdot 341.3333333333333\right), s \cdot 64\right), s \cdot -16\right)\right) \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (*
   u
   (fma
    u
    (*
     u
     (fma
      u
      (fma (* u s) 967.1111111111111 (* s 341.3333333333333))
      (* s 64.0)))
    (* s -16.0)))
  (/ 1.0 (fma u (fma u (fma u 64.0 21.333333333333332) 8.0) -4.0))))

C

float code(float s, float u) {
	return (u * fmaf(u, (u * fmaf(u, fmaf((u * s), 967.1111111111111f, (s * 341.3333333333333f)), (s * 64.0f))), (s * -16.0f))) * (1.0f / fmaf(u, fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f), -4.0f));
}

Julia

function code(s, u)
	return Float32(Float32(u * fma(u, Float32(u * fma(u, fma(Float32(u * s), Float32(967.1111111111111), Float32(s * Float32(341.3333333333333))), Float32(s * Float32(64.0)))), Float32(s * Float32(-16.0)))) * Float32(Float32(1.0) / fma(u, fma(u, fma(u, Float32(64.0), Float32(21.333333333333332)), Float32(8.0)), Float32(-4.0))))
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 93.3

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

5. Simplified 93.3%

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

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

   5. lift-fma.f32 N/A

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

   6. flip-+ N/A

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

   7. associate-*r/ N/A

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

   8. div-inv N/A

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

   9. lower-*.f32 N/A

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

7. Applied egg-rr 92.9%

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

8. Taylor expanded in u around 0

   \[\leadsto \left(\left(u \cdot s\right) \cdot \mathsf{fma}\left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), u \cdot \mathsf{fma}\left(u, \color{blue}{\frac{64}{3}}, 8\right), -16\right)\right) \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]
9. Step-by-step derivation

10. Simplified 93.7%

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

11. Taylor expanded in u around 0

    \[\leadsto \color{blue}{\left(u \cdot \left(-16 \cdot s + {u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]
12. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(u \cdot \left(-16 \cdot s + {u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    2. +-commutative N/A

       \[\leadsto \left(u \cdot \color{blue}{\left({u}^{2} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right) + -16 \cdot s\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    3. unpow2 N/A

       \[\leadsto \left(u \cdot \left(\color{blue}{\left(u \cdot u\right)} \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right) + -16 \cdot s\right)\right) \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    4. associate-*l* N/A

       \[\leadsto \left(u \cdot \left(\color{blue}{u \cdot \left(u \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right)\right)} + -16 \cdot s\right)\right) \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

    5. lower-fma.f32 N/A

       \[\leadsto \left(u \cdot \color{blue}{\mathsf{fma}\left(u, u \cdot \left(64 \cdot s + u \cdot \left(\frac{1024}{3} \cdot s + \frac{8704}{9} \cdot \left(s \cdot u\right)\right)\right), -16 \cdot s\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]

13. Simplified 94.2%

    \[\leadsto \color{blue}{\left(u \cdot \mathsf{fma}\left(u, u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(s \cdot u, 967.1111111111111, s \cdot 341.3333333333333\right), s \cdot 64\right), s \cdot -16\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \]

14. Final simplification 94.2%

    \[\leadsto \left(u \cdot \mathsf{fma}\left(u, u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u \cdot s, 967.1111111111111, s \cdot 341.3333333333333\right), s \cdot 64\right), s \cdot -16\right)\right) \cdot \frac{1}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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))))))