Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 74.1%

Time: 7.3s

Alternatives: 3

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: 61.4% 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: 74.1% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 65.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.1

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

5. Applied rewrites 72.1%

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

6. Final simplification 72.1%

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

Alternative 2: 42.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0034000000450760126:\\ \;\;\;\;\mathsf{fma}\left(s, 0, \left(\left(-8 \cdot u + -4\right) \cdot u\right) \cdot \left(-s\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.0034000000450760126)
   (fma s 0.0 (* (* (+ (* -8.0 u) -4.0) u) (- s)))
   (* (log (/ 1.0 (- 1.0 (* u 4.0)))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.0034000000450760126f) {
		tmp = fmaf(s, 0.0f, ((((-8.0f * u) + -4.0f) * u) * -s));
	} else {
		tmp = logf((1.0f / (1.0f - (u * 4.0f)))) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(u * Float32(4.0)) <= Float32(0.0034000000450760126))
		tmp = fma(s, Float32(0.0), Float32(Float32(Float32(Float32(Float32(-8.0) * u) + Float32(-4.0)) * u) * Float32(-s)));
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(u * Float32(4.0))))) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.00340000005

   1. Initial program 53.0%

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

         \[\leadsto s \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - 4 \cdot u}\right)\right)} \]

      3. metadata-eval N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. metadata-eval 13.6

         \[\leadsto \mathsf{fma}\left(s, \color{blue}{0}, s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)}\right) \]

      9. lift-log.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, 0, s \cdot \color{blue}{\log \left(\frac{1}{1 - 4 \cdot u}\right)}\right) \]

      10. lift-/.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, s \cdot \log \color{blue}{\left(\frac{1}{1 - 4 \cdot u}\right)}\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{fma}\left(s, 0, s \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(1 - 4 \cdot u\right)\right)\right)}\right) \]

      12. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\mathsf{neg}\left(s \cdot \log \left(1 - 4 \cdot u\right)\right)}\right) \]

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

          \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 - 4 \cdot u\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 - 4 \cdot u\right)}\right) \]

      15. lower-neg.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\left(-s\right)} \cdot \log \left(1 - 4 \cdot u\right)\right) \]

      16. lift--.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \log \color{blue}{\left(1 - 4 \cdot u\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(4 \cdot u\right)\right)\right)}\right) \]

      18. lower-log1p.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(4 \cdot u\right)\right)}\right) \]

      19. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{4 \cdot u}\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right) \]

      21. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right) \]

      22. metadata-eval 86.4

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{-4} \cdot u\right)\right) \]

   4. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(-4 \cdot u\right)\right)} \]

   5. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(u \cdot \left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) - 4\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(\left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) - 4\right) \cdot u\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(\left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) - 4\right) \cdot u\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\color{blue}{\left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) + \left(\mathsf{neg}\left(4\right)\right)\right)} \cdot u\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(\color{blue}{\left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) \cdot u} + \left(\mathsf{neg}\left(4\right)\right)\right) \cdot u\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(\left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) \cdot u + \color{blue}{-4}\right) \cdot u\right)\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\color{blue}{\mathsf{fma}\left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8, u, -4\right)} \cdot u\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(-64 \cdot u - \frac{64}{3}\right) + \left(\mathsf{neg}\left(8\right)\right)}, u, -4\right) \cdot u\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(-64 \cdot u - \frac{64}{3}\right) \cdot u} + \left(\mathsf{neg}\left(8\right)\right), u, -4\right) \cdot u\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\left(-64 \cdot u - \frac{64}{3}\right) \cdot u + \color{blue}{-8}, u, -4\right) \cdot u\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-64 \cdot u - \frac{64}{3}, u, -8\right)}, u, -4\right) \cdot u\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-64 \cdot u + \left(\mathsf{neg}\left(\frac{64}{3}\right)\right)}, u, -8\right), u, -4\right) \cdot u\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-64 \cdot u + \color{blue}{\frac{-64}{3}}, u, -8\right), u, -4\right) \cdot u\right)\right) \]

      13. lower-fma.f32 86.4

          \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-64, u, -21.333333333333332\right)}, u, -8\right), u, -4\right) \cdot u\right)\right) \]

   7. Applied rewrites 86.1%

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-64, u, -21.333333333333332\right), u, -8\right), u, -4\right) \cdot u\right)}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.4%

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(-4 + \mathsf{fma}\left(\mathsf{fma}\left(u, -64, -21.333333333333332\right), u, -8\right) \cdot u\right) \cdot u\right)\right) \]

   10. Taylor expanded in u around 0

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(-4 + -8 \cdot u\right) \cdot u\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 98.0%

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(-4 + -8 \cdot u\right) \cdot u\right)\right) \]

   if 0.00340000005 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 92.6%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0034000000450760126:\\ \;\;\;\;\mathsf{fma}\left(s, 0, \left(\left(-8 \cdot u + -4\right) \cdot u\right) \cdot \left(-s\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 22.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(s, 0, \left(\left(-8 \cdot u + -4\right) \cdot u\right) \cdot \left(-s\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (fma s 0.0 (* (* (+ (* -8.0 u) -4.0) u) (- s))))

C

float code(float s, float u) {
	return fmaf(s, 0.0f, ((((-8.0f * u) + -4.0f) * u) * -s));
}

Julia

function code(s, u)
	return fma(s, Float32(0.0), Float32(Float32(Float32(Float32(Float32(-8.0) * u) + Float32(-4.0)) * u) * Float32(-s)))
end

Derivation

1. Initial program 65.6%

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

      \[\leadsto s \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - 4 \cdot u}\right)\right)} \]

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. metadata-eval 12.3

      \[\leadsto \mathsf{fma}\left(s, \color{blue}{0}, s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)}\right) \]

   9. lift-log.f32 N/A

      \[\leadsto \mathsf{fma}\left(s, 0, s \cdot \color{blue}{\log \left(\frac{1}{1 - 4 \cdot u}\right)}\right) \]

   10. lift-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, s \cdot \log \color{blue}{\left(\frac{1}{1 - 4 \cdot u}\right)}\right) \]

   11. log-rec N/A

       \[\leadsto \mathsf{fma}\left(s, 0, s \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(1 - 4 \cdot u\right)\right)\right)}\right) \]

   12. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\mathsf{neg}\left(s \cdot \log \left(1 - 4 \cdot u\right)\right)}\right) \]

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

       \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 - 4 \cdot u\right)}\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(1 - 4 \cdot u\right)}\right) \]

   15. lower-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \color{blue}{\left(-s\right)} \cdot \log \left(1 - 4 \cdot u\right)\right) \]

   16. lift--.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \log \color{blue}{\left(1 - 4 \cdot u\right)}\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \log \color{blue}{\left(1 + \left(\mathsf{neg}\left(4 \cdot u\right)\right)\right)}\right) \]

   18. lower-log1p.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(4 \cdot u\right)\right)}\right) \]

   19. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{4 \cdot u}\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot u}\right)\right) \]

   22. metadata-eval 72.1

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{-4} \cdot u\right)\right) \]

4. Applied rewrites 72.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s, 0, \left(-s\right) \cdot \mathsf{log1p}\left(-4 \cdot u\right)\right)} \]

5. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(u \cdot \left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) - 4\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(\left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) - 4\right) \cdot u\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(\left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) - 4\right) \cdot u\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\color{blue}{\left(u \cdot \left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) + \left(\mathsf{neg}\left(4\right)\right)\right)} \cdot u\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(\color{blue}{\left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) \cdot u} + \left(\mathsf{neg}\left(4\right)\right)\right) \cdot u\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(\left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8\right) \cdot u + \color{blue}{-4}\right) \cdot u\right)\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\color{blue}{\mathsf{fma}\left(u \cdot \left(-64 \cdot u - \frac{64}{3}\right) - 8, u, -4\right)} \cdot u\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(-64 \cdot u - \frac{64}{3}\right) + \left(\mathsf{neg}\left(8\right)\right)}, u, -4\right) \cdot u\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(-64 \cdot u - \frac{64}{3}\right) \cdot u} + \left(\mathsf{neg}\left(8\right)\right), u, -4\right) \cdot u\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\left(-64 \cdot u - \frac{64}{3}\right) \cdot u + \color{blue}{-8}, u, -4\right) \cdot u\right)\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-64 \cdot u - \frac{64}{3}, u, -8\right)}, u, -4\right) \cdot u\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-64 \cdot u + \left(\mathsf{neg}\left(\frac{64}{3}\right)\right)}, u, -8\right), u, -4\right) \cdot u\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-64 \cdot u + \color{blue}{\frac{-64}{3}}, u, -8\right), u, -4\right) \cdot u\right)\right) \]

   13. lower-fma.f32 72.1

       \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-64, u, -21.333333333333332\right)}, u, -8\right), u, -4\right) \cdot u\right)\right) \]

7. Applied rewrites 72.0%

   \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-64, u, -21.333333333333332\right), u, -8\right), u, -4\right) \cdot u\right)}\right) \]
8. Step-by-step derivation

9. Applied rewrites 84.2%

   \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(-4 + \mathsf{fma}\left(\mathsf{fma}\left(u, -64, -21.333333333333332\right), u, -8\right) \cdot u\right) \cdot u\right)\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(-4 + -8 \cdot u\right) \cdot u\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 84.3%

    \[\leadsto \mathsf{fma}\left(s, 0, \left(-s\right) \cdot \left(\left(-4 + -8 \cdot u\right) \cdot u\right)\right) \]

13. Final simplification 54.5%

    \[\leadsto \mathsf{fma}\left(s, 0, \left(\left(-8 \cdot u + -4\right) \cdot u\right) \cdot \left(-s\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024284 
(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))))))