Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.0% → 96.8%

Time: 6.0s

Alternatives: 5

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: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.006200000178068876:\\ \;\;\;\;\frac{-1}{0.5 - \frac{0.25}{u}} \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.006200000178068876)
   (* (/ -1.0 (- 0.5 (/ 0.25 u))) s)
   (* (log (/ 1.0 (- 1.0 (exp (log (* u 4.0)))))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.006200000178068876f) {
		tmp = (-1.0f / (0.5f - (0.25f / u))) * s;
	} else {
		tmp = logf((1.0f / (1.0f - expf(logf((u * 4.0f)))))) * s;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((u * 4.0e0) <= 0.006200000178068876e0) then
        tmp = ((-1.0e0) / (0.5e0 - (0.25e0 / u))) * s
    else
        tmp = log((1.0e0 / (1.0e0 - exp(log((u * 4.0e0)))))) * s
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(u * Float32(4.0)) <= Float32(0.006200000178068876))
		tmp = Float32(Float32(Float32(-1.0) / Float32(Float32(0.5) - Float32(Float32(0.25) / u))) * s);
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - exp(log(Float32(u * Float32(4.0))))))) * s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((u * single(4.0)) <= single(0.006200000178068876))
		tmp = (single(-1.0) / (single(0.5) - (single(0.25) / u))) * s;
	else
		tmp = log((single(1.0) / (single(1.0) - exp(log((u * single(4.0))))))) * s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.00620000018

   1. Initial program 47.9%

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

   4. Applied rewrites 70.1%

      \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{s \cdot \frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      2. lift-/.f32 N/A

         \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{s}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}\right)}} \]

      5. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\color{blue}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}\right)} \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{\frac{\mathsf{neg}\left(\left(-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      7. lift-neg.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

      9. lift-pow.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

      10. lift-pow.f32 N/A

          \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      11. pow-div N/A

          \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\left(2 - 3\right)}}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(s\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\color{blue}{-1}}} \]

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

   6. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\frac{-s}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}}} \]

   7. Taylor expanded in u around 0

      \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u - \frac{1}{4}}{u}}} \]
   8. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u}{u} - \frac{\frac{1}{4}}{u}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} \cdot \frac{u}{u}} - \frac{\frac{1}{4}}{u}} \]

      3. *-inverses N/A

         \[\leadsto \frac{-s}{\frac{1}{2} \cdot \color{blue}{1} - \frac{\frac{1}{4}}{u}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2}} - \frac{\frac{1}{4}}{u}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4} \cdot 1}}{u}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{1}{4} \cdot \frac{1}{u}}} \]

      7. lower--.f32 N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{u}}} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{\frac{1}{4} \cdot 1}{u}}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4}}}{u}} \]

      10. lower-/.f32 98.2

          \[\leadsto \frac{-s}{0.5 - \color{blue}{\frac{0.25}{u}}} \]

   9. Applied rewrites 98.2%

      \[\leadsto \frac{-s}{\color{blue}{0.5 - \frac{0.25}{u}}} \]
   10. Step-by-step derivation

       1. lift-/.f32 N/A

          \[\leadsto \color{blue}{\frac{-s}{\frac{1}{2} - \frac{\frac{1}{4}}{u}}} \]

       2. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(-s\right)\right)}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

       3. div-inv N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-s\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

       4. lift-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)} \]

       5. remove-double-neg N/A

          \[\leadsto \color{blue}{s} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)} \]

       6. lower-*.f32 N/A

          \[\leadsto \color{blue}{s \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

       7. frac-2neg N/A

          \[\leadsto s \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)\right)\right)}} \]

       8. metadata-eval N/A

          \[\leadsto s \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)\right)\right)} \]

       9. remove-double-neg N/A

          \[\leadsto s \cdot \frac{-1}{\color{blue}{\frac{1}{2} - \frac{\frac{1}{4}}{u}}} \]

       10. lower-/.f32 98.2

           \[\leadsto s \cdot \color{blue}{\frac{-1}{0.5 - \frac{0.25}{u}}} \]

   11. Applied rewrites 98.2%

       \[\leadsto \color{blue}{s \cdot \frac{-1}{0.5 - \frac{0.25}{u}}} \]

   if 0.00620000018 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 93.2%

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

      1. rem-exp-log N/A

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

      2. lower-exp.f32 N/A

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

      3. lower-log.f32 93.3

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

      \[\leadsto s \cdot \log \left(\frac{1}{1 - \color{blue}{e^{\log \left(u \cdot 4\right)}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.006200000178068876:\\ \;\;\;\;\frac{-1}{0.5 - \frac{0.25}{u}} \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.9936000108718872:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.5 - \frac{0.25}{u}} \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.9936000108718872)
     (* (log (/ 1.0 t_0)) s)
     (* (/ -1.0 (- 0.5 (/ 0.25 u))) s))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (u * 4.0f);
	float tmp;
	if (t_0 <= 0.9936000108718872f) {
		tmp = logf((1.0f / t_0)) * s;
	} else {
		tmp = (-1.0f / (0.5f - (0.25f / u))) * s;
	}
	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 <= 0.9936000108718872e0) then
        tmp = log((1.0e0 / t_0)) * s
    else
        tmp = ((-1.0e0) / (0.5e0 - (0.25e0 / u))) * s
    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(0.9936000108718872))
		tmp = Float32(log(Float32(Float32(1.0) / t_0)) * s);
	else
		tmp = Float32(Float32(Float32(-1.0) / Float32(Float32(0.5) - Float32(Float32(0.25) / u))) * s);
	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(0.9936000108718872))
		tmp = log((single(1.0) / t_0)) * s;
	else
		tmp = (single(-1.0) / (single(0.5) - (single(0.25) / u))) * s;
	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)) < 0.993600011

   1. Initial program 93.2%

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

   if 0.993600011 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 47.9%

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

   4. Applied rewrites 71.0%

      \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{s \cdot \frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      2. lift-/.f32 N/A

         \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{s}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}\right)}} \]

      5. lift-/.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\color{blue}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}\right)} \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{\frac{\mathsf{neg}\left(\left(-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      7. lift-neg.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

      8. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

      9. lift-pow.f32 N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

      10. lift-pow.f32 N/A

          \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

      11. pow-div N/A

          \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\left(2 - 3\right)}}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(s\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\color{blue}{-1}}} \]

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

   6. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\frac{-s}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}}} \]

   7. Taylor expanded in u around 0

      \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u - \frac{1}{4}}{u}}} \]
   8. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u}{u} - \frac{\frac{1}{4}}{u}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} \cdot \frac{u}{u}} - \frac{\frac{1}{4}}{u}} \]

      3. *-inverses N/A

         \[\leadsto \frac{-s}{\frac{1}{2} \cdot \color{blue}{1} - \frac{\frac{1}{4}}{u}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2}} - \frac{\frac{1}{4}}{u}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4} \cdot 1}}{u}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{1}{4} \cdot \frac{1}{u}}} \]

      7. lower--.f32 N/A

         \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{u}}} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{\frac{1}{4} \cdot 1}{u}}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4}}}{u}} \]

      10. lower-/.f32 98.2

          \[\leadsto \frac{-s}{0.5 - \color{blue}{\frac{0.25}{u}}} \]

   9. Applied rewrites 98.2%

      \[\leadsto \frac{-s}{\color{blue}{0.5 - \frac{0.25}{u}}} \]
   10. Step-by-step derivation

       1. lift-/.f32 N/A

          \[\leadsto \color{blue}{\frac{-s}{\frac{1}{2} - \frac{\frac{1}{4}}{u}}} \]

       2. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(-s\right)\right)}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

       3. div-inv N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-s\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

       4. lift-neg.f32 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)} \]

       5. remove-double-neg N/A

          \[\leadsto \color{blue}{s} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)} \]

       6. lower-*.f32 N/A

          \[\leadsto \color{blue}{s \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

       7. frac-2neg N/A

          \[\leadsto s \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)\right)\right)}} \]

       8. metadata-eval N/A

          \[\leadsto s \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)\right)\right)} \]

       9. remove-double-neg N/A

          \[\leadsto s \cdot \frac{-1}{\color{blue}{\frac{1}{2} - \frac{\frac{1}{4}}{u}}} \]

       10. lower-/.f32 98.2

           \[\leadsto s \cdot \color{blue}{\frac{-1}{0.5 - \frac{0.25}{u}}} \]

   11. Applied rewrites 98.2%

       \[\leadsto \color{blue}{s \cdot \frac{-1}{0.5 - \frac{0.25}{u}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9936000108718872:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{0.5 - \frac{0.25}{u}} \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{0.5 - \frac{0.25}{u}} \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (/ -1.0 (- 0.5 (/ 0.25 u))) s))

C

float code(float s, float u) {
	return (-1.0f / (0.5f - (0.25f / u))) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-1.0e0) / (0.5e0 - (0.25e0 / u))) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-1.0) / Float32(Float32(0.5) - Float32(Float32(0.25) / u))) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (single(-1.0) / (single(0.5) - (single(0.25) / u))) * s;
end

Derivation

1. Initial program 58.4%

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

4. Applied rewrites 53.0%

   \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{s \cdot \frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{s}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   4. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}\right)}} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\color{blue}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}\right)} \]

   6. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{\frac{\mathsf{neg}\left(\left(-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   7. lift-neg.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

   8. remove-double-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

   9. lift-pow.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

   10. lift-pow.f32 N/A

       \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   11. pow-div N/A

       \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\left(2 - 3\right)}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(s\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\color{blue}{-1}}} \]

   13. metadata-eval N/A

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

   14. lower-/.f32 N/A

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

6. Applied rewrites 74.2%

   \[\leadsto \color{blue}{\frac{-s}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}}} \]

7. Taylor expanded in u around 0

   \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u - \frac{1}{4}}{u}}} \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u}{u} - \frac{\frac{1}{4}}{u}}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} \cdot \frac{u}{u}} - \frac{\frac{1}{4}}{u}} \]

   3. *-inverses N/A

      \[\leadsto \frac{-s}{\frac{1}{2} \cdot \color{blue}{1} - \frac{\frac{1}{4}}{u}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2}} - \frac{\frac{1}{4}}{u}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4} \cdot 1}}{u}} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{1}{4} \cdot \frac{1}{u}}} \]

   7. lower--.f32 N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{u}}} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{\frac{1}{4} \cdot 1}{u}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4}}}{u}} \]

   10. lower-/.f32 88.8

       \[\leadsto \frac{-s}{0.5 - \color{blue}{\frac{0.25}{u}}} \]

9. Applied rewrites 88.8%

   \[\leadsto \frac{-s}{\color{blue}{0.5 - \frac{0.25}{u}}} \]
10. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto \color{blue}{\frac{-s}{\frac{1}{2} - \frac{\frac{1}{4}}{u}}} \]

    2. frac-2neg N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(-s\right)\right)}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

    3. div-inv N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-s\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

    4. lift-neg.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)} \]

    5. remove-double-neg N/A

       \[\leadsto \color{blue}{s} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)} \]

    6. lower-*.f32 N/A

       \[\leadsto \color{blue}{s \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)}} \]

    7. frac-2neg N/A

       \[\leadsto s \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)\right)\right)}} \]

    8. metadata-eval N/A

       \[\leadsto s \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} - \frac{\frac{1}{4}}{u}\right)\right)\right)\right)} \]

    9. remove-double-neg N/A

       \[\leadsto s \cdot \frac{-1}{\color{blue}{\frac{1}{2} - \frac{\frac{1}{4}}{u}}} \]

    10. lower-/.f32 88.8

        \[\leadsto s \cdot \color{blue}{\frac{-1}{0.5 - \frac{0.25}{u}}} \]

11. Applied rewrites 88.8%

    \[\leadsto \color{blue}{s \cdot \frac{-1}{0.5 - \frac{0.25}{u}}} \]

12. Final simplification 88.8%

    \[\leadsto \frac{-1}{0.5 - \frac{0.25}{u}} \cdot s \]
13. Add Preprocessing

Alternative 4: 88.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{-s}{0.5 - \frac{0.25}{u}} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ (- s) (- 0.5 (/ 0.25 u))))

C

float code(float s, float u) {
	return -s / (0.5f - (0.25f / u));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = -s / (0.5e0 - (0.25e0 / u))
end function

Julia

function code(s, u)
	return Float32(Float32(-s) / Float32(Float32(0.5) - Float32(Float32(0.25) / u)))
end

MATLAB

function tmp = code(s, u)
	tmp = -s / (single(0.5) - (single(0.25) / u));
end

Derivation

1. Initial program 58.4%

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

4. Applied rewrites 52.8%

   \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{s \cdot \frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto s \cdot \color{blue}{\frac{1}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{s}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   4. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}\right)}} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\mathsf{neg}\left(\color{blue}{\frac{-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}\right)} \]

   6. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{\frac{\mathsf{neg}\left(\left(-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   7. lift-neg.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right)\right)}\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

   8. remove-double-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

   9. lift-pow.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}} \]

   10. lift-pow.f32 N/A

       \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\frac{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{3}}}} \]

   11. pow-div N/A

       \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\left(2 - 3\right)}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{neg}\left(s\right)}{{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{\color{blue}{-1}}} \]

   13. metadata-eval N/A

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

   14. lower-/.f32 N/A

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

6. Applied rewrites 74.0%

   \[\leadsto \color{blue}{\frac{-s}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}}} \]

7. Taylor expanded in u around 0

   \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u - \frac{1}{4}}{u}}} \]
8. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{\frac{1}{2} \cdot u}{u} - \frac{\frac{1}{4}}{u}}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} \cdot \frac{u}{u}} - \frac{\frac{1}{4}}{u}} \]

   3. *-inverses N/A

      \[\leadsto \frac{-s}{\frac{1}{2} \cdot \color{blue}{1} - \frac{\frac{1}{4}}{u}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2}} - \frac{\frac{1}{4}}{u}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4} \cdot 1}}{u}} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{1}{4} \cdot \frac{1}{u}}} \]

   7. lower--.f32 N/A

      \[\leadsto \frac{-s}{\color{blue}{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{u}}} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \color{blue}{\frac{\frac{1}{4} \cdot 1}{u}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{-s}{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4}}}{u}} \]

   10. lower-/.f32 88.8

       \[\leadsto \frac{-s}{0.5 - \color{blue}{\frac{0.25}{u}}} \]

9. Applied rewrites 88.8%

   \[\leadsto \frac{-s}{\color{blue}{0.5 - \frac{0.25}{u}}} \]
10. Add Preprocessing

Alternative 5: 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 58.4%

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

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

5. Applied rewrites 74.4%

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

6. Final simplification 74.4%

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

Reproduce

herbie shell --seed 2024308 
(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))))))