Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.0% → 97.9%

Time: 10.8s

Alternatives: 6

Speedup: 11.4×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \leq 0.004000000189989805:\\ \;\;\;\;\left(-s\right) \cdot \left(\left(\left(-21.333333333333332 \cdot u - 8\right) \cdot u - 4\right) \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= u 0.004000000189989805)
   (* (- s) (* (- (* (- (* -21.333333333333332 u) 8.0) u) 4.0) u))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if (u <= 0.004000000189989805f) {
		tmp = -s * (((((-21.333333333333332f * u) - 8.0f) * u) - 4.0f) * u);
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if (u <= 0.004000000189989805e0) then
        tmp = -s * ((((((-21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u)
    else
        tmp = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (u <= Float32(0.004000000189989805))
		tmp = Float32(Float32(-s) * Float32(Float32(Float32(Float32(Float32(Float32(-21.333333333333332) * u) - Float32(8.0)) * u) - Float32(4.0)) * u));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if (u <= single(0.004000000189989805))
		tmp = -s * (((((single(-21.333333333333332) * u) - single(8.0)) * u) - single(4.0)) * u);
	else
		tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u < 0.00400000019

   1. Initial program 53.3%

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

   3. Taylor expanded in u around 0

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

      1. lower-*.f32 81.5

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in u around 0

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

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 81.5

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

   10. Applied rewrites 81.3%

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

   11. Taylor expanded in s around -inf

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

   13. Applied rewrites 98.8%

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

   if 0.00400000019 < u

   1. Initial program 94.3%

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

Alternative 2: 91.4% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = -s * ((((((-21.333333333333332e0) * u) - 8.0e0) * u) - 4.0e0) * u)
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = -s * (((((single(-21.333333333333332) * u) - single(8.0)) * u) - single(4.0)) * u);
end

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 73.1

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 72.8%

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

8. Taylor expanded in u around 0

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

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 73.1

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

10. Applied rewrites 72.9%

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

11. Taylor expanded in s around -inf

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

13. Applied rewrites 91.2%

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

Alternative 3: 87.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (((8.0e0 * u) * u) + (4.0e0 * u))
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * (((single(8.0) * u) * u) + (single(4.0) * u));
end

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 73.1

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 72.8%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.1

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

10. Applied rewrites 72.9%

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

12. Applied rewrites 86.7%

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

Alternative 4: 87.1% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((4.0e0 - ((-8.0e0) * u)) * u)
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * ((single(4.0) - (single(-8.0) * u)) * u);
end

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 73.1

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 72.8%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.1

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

10. Applied rewrites 72.9%

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

12. Applied rewrites 86.6%

    \[\leadsto s \cdot \left(\left(4 - -8 \cdot u\right) \cdot u\right) \]
13. Add Preprocessing

Alternative 5: 74.2% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 73.1

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

5. Applied rewrites 73.1%

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

Alternative 6: 74.0% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 73.1

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

5. Applied rewrites 73.1%

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

7. Applied rewrites 72.8%

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

8. Taylor expanded in u around 0

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

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 73.1

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

10. Applied rewrites 72.9%

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

11. Taylor expanded in u around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 72.8

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

13. Applied rewrites 72.8%

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

Reproduce

herbie shell --seed 2024337 
(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))))))