Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 98.4%

Time: 7.9s

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: 61.2% 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: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9599999785423279:\\ \;\;\;\;\log \left(\frac{1}{1 - \frac{1}{\frac{0.25}{u}}}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{21.333333333333332}{u} + \left(\frac{8}{u \cdot u} + 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right) \cdot s\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* u 4.0)) 0.9599999785423279)
   (* (log (/ 1.0 (- 1.0 (/ 1.0 (/ 0.25 u))))) s)
   (*
    (*
     (+
      (* (* (+ (/ 21.333333333333332 u) (+ (/ 8.0 (* u u)) 64.0)) (* u u)) u)
      4.0)
     s)
    u)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (u * 4.0f)) <= 0.9599999785423279f) {
		tmp = logf((1.0f / (1.0f - (1.0f / (0.25f / u))))) * s;
	} else {
		tmp = ((((((21.333333333333332f / u) + ((8.0f / (u * u)) + 64.0f)) * (u * u)) * u) + 4.0f) * s) * u;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((1.0e0 - (u * 4.0e0)) <= 0.9599999785423279e0) then
        tmp = log((1.0e0 / (1.0e0 - (1.0e0 / (0.25e0 / u))))) * s
    else
        tmp = ((((((21.333333333333332e0 / u) + ((8.0e0 / (u * u)) + 64.0e0)) * (u * u)) * u) + 4.0e0) * s) * u
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(u * Float32(4.0))) <= Float32(0.9599999785423279))
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(1.0) / Float32(Float32(0.25) / u))))) * s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(21.333333333333332) / u) + Float32(Float32(Float32(8.0) / Float32(u * u)) + Float32(64.0))) * Float32(u * u)) * u) + Float32(4.0)) * s) * u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(1.0) - (u * single(4.0))) <= single(0.9599999785423279))
		tmp = log((single(1.0) / (single(1.0) - (single(1.0) / (single(0.25) / u))))) * s;
	else
		tmp = ((((((single(21.333333333333332) / u) + ((single(8.0) / (u * u)) + single(64.0))) * (u * u)) * u) + single(4.0)) * 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)) < 0.959999979

   1. Initial program 95.3%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. lift-*.f32 N/A

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

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

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

      9. lower-*.f32 N/A

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

      10. metadata-eval 95.4

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

   4. Applied rewrites 95.4%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval 95.4

         \[\leadsto s \cdot \log \left(\frac{1}{1 - \frac{1}{\frac{\color{blue}{0.25}}{u}}}\right) \]

   6. Applied rewrites 95.4%

      \[\leadsto s \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{1}{\frac{0.25}{u}}}}\right) \]

   if 0.959999979 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 56.8%

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

   3. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \left(s \cdot \left(\left(\left(\left(\frac{8}{u \cdot u} + 64\right) + \frac{21.333333333333332}{u}\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right)\right) \cdot u \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9599999785423279:\\ \;\;\;\;\log \left(\frac{1}{1 - \frac{1}{\frac{0.25}{u}}}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{21.333333333333332}{u} + \left(\frac{8}{u \cdot u} + 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right) \cdot s\right) \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.9599999785423279:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{21.333333333333332}{u} + \left(\frac{8}{u \cdot u} + 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right) \cdot s\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.9599999785423279)
     (* (log (/ 1.0 t_0)) s)
     (*
      (*
       (+
        (* (* (+ (/ 21.333333333333332 u) (+ (/ 8.0 (* u u)) 64.0)) (* u u)) u)
        4.0)
       s)
      u))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (u * 4.0f);
	float tmp;
	if (t_0 <= 0.9599999785423279f) {
		tmp = logf((1.0f / t_0)) * s;
	} else {
		tmp = ((((((21.333333333333332f / u) + ((8.0f / (u * u)) + 64.0f)) * (u * u)) * u) + 4.0f) * 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 <= 0.9599999785423279e0) then
        tmp = log((1.0e0 / t_0)) * s
    else
        tmp = ((((((21.333333333333332e0 / u) + ((8.0e0 / (u * u)) + 64.0e0)) * (u * u)) * u) + 4.0e0) * 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(0.9599999785423279))
		tmp = Float32(log(Float32(Float32(1.0) / t_0)) * s);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(21.333333333333332) / u) + Float32(Float32(Float32(8.0) / Float32(u * u)) + Float32(64.0))) * Float32(u * u)) * u) + Float32(4.0)) * 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(0.9599999785423279))
		tmp = log((single(1.0) / t_0)) * s;
	else
		tmp = ((((((single(21.333333333333332) / u) + ((single(8.0) / (u * u)) + single(64.0))) * (u * u)) * u) + single(4.0)) * 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)) < 0.959999979

   1. Initial program 95.3%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 56.8%

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

   3. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \left(s \cdot \left(\left(\left(\left(\frac{8}{u \cdot u} + 64\right) + \frac{21.333333333333332}{u}\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right)\right) \cdot u \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9599999785423279:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{21.333333333333332}{u} + \left(\frac{8}{u \cdot u} + 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right) \cdot s\right) \cdot u\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((((((21.333333333333332e0 / u) + ((8.0e0 / (u * u)) + 64.0e0)) * (u * u)) * u) + 4.0e0) * s) * u
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = ((((((single(21.333333333333332) / u) + ((single(8.0) / (u * u)) + single(64.0))) * (u * u)) * u) + single(4.0)) * s) * u;
end

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 74.4%

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

7. Applied rewrites 88.1%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 93.9%

    \[\leadsto \left(s \cdot \left(\left(\left(\left(\frac{8}{u \cdot u} + 64\right) + \frac{21.333333333333332}{u}\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right)\right) \cdot u \]

11. Final simplification 93.9%

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

Alternative 4: 86.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. log-div N/A

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

   6. flip-- N/A

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

   7. metadata-eval N/A

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

   8. +-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 43.4%

   \[\leadsto \color{blue}{\frac{\left(-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right) \cdot s}{\mathsf{log1p}\left(-4 \cdot u\right)}} \]

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 74.4

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

7. Applied rewrites 74.4%

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

9. Applied rewrites 88.4%

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

10. Final simplification 88.4%

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

Alternative 5: 86.7% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. log-div N/A

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

   6. flip-- N/A

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

   7. metadata-eval N/A

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

   8. +-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 43.4%

   \[\leadsto \color{blue}{\frac{\left(-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right) \cdot s}{\mathsf{log1p}\left(-4 \cdot u\right)}} \]

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 74.4

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

7. Applied rewrites 74.4%

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

9. Applied rewrites 88.4%

   \[\leadsto \left(\left(s \cdot u\right) \cdot 8 + s \cdot 4\right) \cdot u \]

10. Final simplification 88.4%

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

Alternative 6: 86.5% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 74.4%

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

7. Applied rewrites 88.2%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 88.2%

    \[\leadsto \left(s \cdot \left(8 \cdot u + 4\right)\right) \cdot u \]

11. Final simplification 88.2%

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

Alternative 7: 73.9% 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 62.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 74.4

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

5. Applied rewrites 74.4%

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

6. Final simplification 74.4%

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

Reproduce

herbie shell --seed 2024250 
(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))))))