Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 97.7%

Time: 7.7s

Alternatives: 7

Speedup: 11.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.9800000190734863:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{4}{u} + 8}{u} \cdot u - -21.333333333333332 \cdot u\right) \cdot u\right) \cdot s\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.9800000190734863)
     (* (log (/ 1.0 t_0)) s)
     (*
      (* (* (- (* (/ (+ (/ 4.0 u) 8.0) u) u) (* -21.333333333333332 u)) u) s)
      u))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (u * 4.0f);
	float tmp;
	if (t_0 <= 0.9800000190734863f) {
		tmp = logf((1.0f / t_0)) * s;
	} else {
		tmp = (((((((4.0f / u) + 8.0f) / u) * u) - (-21.333333333333332f * u)) * u) * s) * u;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 - (u * 4.0e0)
    if (t_0 <= 0.9800000190734863e0) then
        tmp = log((1.0e0 / t_0)) * s
    else
        tmp = (((((((4.0e0 / u) + 8.0e0) / u) * u) - ((-21.333333333333332e0) * u)) * u) * s) * u
    end if
    code = tmp
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(u * Float32(4.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9800000190734863))
		tmp = Float32(log(Float32(Float32(1.0) / t_0)) * s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(4.0) / u) + Float32(8.0)) / u) * u) - Float32(Float32(-21.333333333333332) * u)) * u) * s) * u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	t_0 = single(1.0) - (u * single(4.0));
	tmp = single(0.0);
	if (t_0 <= single(0.9800000190734863))
		tmp = log((single(1.0) / t_0)) * s;
	else
		tmp = (((((((single(4.0) / u) + single(8.0)) / u) * u) - (single(-21.333333333333332) * u)) * u) * s) * u;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.980000019

   1. Initial program 95.2%

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

   if 0.980000019 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 53.1%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in u around inf

      \[\leadsto \left(s \cdot \left({u}^{2} \cdot \left(\frac{64}{3} + \left(8 \cdot \frac{1}{u} + \frac{4}{{u}^{2}}\right)\right)\right)\right) \cdot u \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto \left(s \cdot \left(\left(\left(\left(21.333333333333332 + \frac{4}{u \cdot u}\right) + \frac{8}{u}\right) \cdot u\right) \cdot u\right)\right) \cdot u \]

   9. Taylor expanded in u around -inf

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

   11. Applied rewrites 98.4%

       \[\leadsto \left(s \cdot \left(\left(\left(-u\right) \cdot \left(-21.333333333333332 - \frac{\frac{4}{u} + 8}{u}\right)\right) \cdot u\right)\right) \cdot u \]
   12. Step-by-step derivation

   13. Applied rewrites 98.5%

       \[\leadsto \left(s \cdot \left(\left(\left(-u\right) \cdot -21.333333333333332 + \left(-u\right) \cdot \frac{-\left(8 + \frac{4}{u}\right)}{u}\right) \cdot u\right)\right) \cdot u \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9800000190734863:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{4}{u} + 8}{u} \cdot u - -21.333333333333332 \cdot u\right) \cdot u\right) \cdot s\right) \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9999899864196777:\\ \;\;\;\;\left(\frac{u}{\mathsf{fma}\left(21.333333333333332, u, -8\right)} \cdot \mathsf{fma}\left(u \cdot u, 455.1111111111111, -64\right) - -4\right) \cdot \left(s \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* u 4.0)) 0.9999899864196777)
   (*
    (-
     (*
      (/ u (fma 21.333333333333332 u -8.0))
      (fma (* u u) 455.1111111111111 -64.0))
     -4.0)
    (* s u))
   (* (* u 4.0) s)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (u * 4.0f)) <= 0.9999899864196777f) {
		tmp = (((u / fmaf(21.333333333333332f, u, -8.0f)) * fmaf((u * u), 455.1111111111111f, -64.0f)) - -4.0f) * (s * u);
	} else {
		tmp = (u * 4.0f) * s;
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(u * Float32(4.0))) <= Float32(0.9999899864196777))
		tmp = Float32(Float32(Float32(Float32(u / fma(Float32(21.333333333333332), u, Float32(-8.0))) * fma(Float32(u * u), Float32(455.1111111111111), Float32(-64.0))) - Float32(-4.0)) * Float32(s * u));
	else
		tmp = Float32(Float32(u * Float32(4.0)) * s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.999989986

   1. Initial program 79.9%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 55.5%

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

   9. Applied rewrites 55.5%

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

   10. Taylor expanded in s around -inf

       \[\leadsto -1 \cdot \color{blue}{\left(s \cdot \left(u \cdot \left(-1 \cdot \frac{u \cdot \left(\frac{4096}{9} \cdot {u}^{2} - 64\right)}{\frac{64}{3} \cdot u - 8} - 4\right)\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 75.4%

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

   if 0.999989986 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 38.3%

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

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

      2. lower-*.f32 95.1

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

   5. Applied rewrites 95.1%

      \[\leadsto s \cdot \color{blue}{\left(u \cdot 4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9999899864196777:\\ \;\;\;\;\left(\frac{u}{\mathsf{fma}\left(21.333333333333332, u, -8\right)} \cdot \mathsf{fma}\left(u \cdot u, 455.1111111111111, -64\right) - -4\right) \cdot \left(s \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.0

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

5. Applied rewrites 73.0%

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

6. Taylor expanded in u around inf

   \[\leadsto \left(s \cdot \left({u}^{2} \cdot \left(\frac{64}{3} + \left(8 \cdot \frac{1}{u} + \frac{4}{{u}^{2}}\right)\right)\right)\right) \cdot u \]
7. Step-by-step derivation

8. Applied rewrites 89.9%

   \[\leadsto \left(s \cdot \left(\left(\left(\left(21.333333333333332 + \frac{4}{u \cdot u}\right) + \frac{8}{u}\right) \cdot u\right) \cdot u\right)\right) \cdot u \]

9. Taylor expanded in u around -inf

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

11. Applied rewrites 90.1%

    \[\leadsto \left(s \cdot \left(\left(\left(-u\right) \cdot \left(-21.333333333333332 - \frac{\frac{4}{u} + 8}{u}\right)\right) \cdot u\right)\right) \cdot u \]
12. Step-by-step derivation

13. Applied rewrites 90.1%

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

14. Final simplification 90.1%

    \[\leadsto \left(\left(\left(\frac{\frac{4}{u} + 8}{u} \cdot u - -21.333333333333332 \cdot u\right) \cdot u\right) \cdot s\right) \cdot u \]
15. Add Preprocessing

Alternative 4: 90.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.0

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

5. Applied rewrites 73.0%

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

6. Taylor expanded in u around inf

   \[\leadsto \left(s \cdot \left({u}^{2} \cdot \left(\frac{64}{3} + \left(8 \cdot \frac{1}{u} + \frac{4}{{u}^{2}}\right)\right)\right)\right) \cdot u \]
7. Step-by-step derivation

8. Applied rewrites 89.9%

   \[\leadsto \left(s \cdot \left(\left(\left(\left(21.333333333333332 + \frac{4}{u \cdot u}\right) + \frac{8}{u}\right) \cdot u\right) \cdot u\right)\right) \cdot u \]

9. Taylor expanded in u around -inf

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

11. Applied rewrites 90.1%

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

12. Final simplification 90.1%

    \[\leadsto \left(\left(\left(\left(\frac{\frac{4}{u} + 8}{u} - -21.333333333333332\right) \cdot u\right) \cdot u\right) \cdot s\right) \cdot u \]
13. Add Preprocessing

Alternative 5: 90.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.0

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

5. Applied rewrites 73.0%

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

6. Taylor expanded in u around inf

   \[\leadsto \left(s \cdot \left({u}^{2} \cdot \left(\frac{64}{3} + \left(8 \cdot \frac{1}{u} + \frac{4}{{u}^{2}}\right)\right)\right)\right) \cdot u \]
7. Step-by-step derivation

8. Applied rewrites 89.9%

   \[\leadsto \left(s \cdot \left(\left(\left(\left(21.333333333333332 + \frac{4}{u \cdot u}\right) + \frac{8}{u}\right) \cdot u\right) \cdot u\right)\right) \cdot u \]

9. Taylor expanded in u around -inf

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

11. Applied rewrites 90.1%

    \[\leadsto \left(s \cdot \left(\left(\left(-u\right) \cdot \left(-21.333333333333332 - \frac{\frac{4}{u} + 8}{u}\right)\right) \cdot u\right)\right) \cdot u \]
12. Step-by-step derivation

13. Applied rewrites 89.8%

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

14. Final simplification 89.8%

    \[\leadsto \left(\left(\left(\frac{\frac{4}{u} + 8}{u} - -21.333333333333332\right) \cdot u\right) \cdot u\right) \cdot \left(s \cdot u\right) \]
15. Add Preprocessing

Alternative 6: 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 61.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.0

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

5. Applied rewrites 73.0%

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

6. Final simplification 73.0%

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

Alternative 7: 73.9% 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 61.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.0

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

5. Applied rewrites 73.0%

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

7. Applied rewrites 72.8%

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

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 72.8%

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

Reproduce

herbie shell --seed 2024255 
(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))))))