Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 88.9%

Time: 7.1s

Alternatives: 4

Speedup: 11.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 88.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9998599886894226:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9998599886894226)
   (* s (log (/ 1.0 (- 1.0 (exp (log (* u 4.0)))))))
   (* s (* 4.0 u))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9998599886894226f) {
		tmp = s * logf((1.0f / (1.0f - expf(logf((u * 4.0f))))));
	} else {
		tmp = s * (4.0f * u);
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((1.0e0 - (4.0e0 * u)) <= 0.9998599886894226e0) then
        tmp = s * log((1.0e0 / (1.0e0 - exp(log((u * 4.0e0))))))
    else
        tmp = s * (4.0e0 * u)
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9998599886894226))
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - exp(log(Float32(u * Float32(4.0))))))));
	else
		tmp = Float32(s * Float32(Float32(4.0) * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(1.0) - (single(4.0) * u)) <= single(0.9998599886894226))
		tmp = s * log((single(1.0) / (single(1.0) - exp(log((u * single(4.0)))))));
	else
		tmp = s * (single(4.0) * 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)) < 0.999859989

   1. Initial program 84.3%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 84.3

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

      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 84.3

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

   4. Applied rewrites 84.3%

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

   if 0.999859989 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 45.0%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

Alternative 2: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot \frac{-1}{s}}{0.5 - \frac{0.25}{u}}\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))) 2.0000000390829628e-24)
   (* s (* 4.0 u))
   (/ (* (* s s) (/ -1.0 s)) (- 0.5 (/ 0.25 u)))))

C

float code(float s, float u) {
	float tmp;
	if ((s * logf((1.0f / (1.0f - (4.0f * u))))) <= 2.0000000390829628e-24f) {
		tmp = s * (4.0f * u);
	} else {
		tmp = ((s * s) * (-1.0f / s)) / (0.5f - (0.25f / u));
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))) <= 2.0000000390829628e-24) then
        tmp = s * (4.0e0 * u)
    else
        tmp = ((s * s) * ((-1.0e0) / s)) / (0.5e0 - (0.25e0 / u))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u))))) <= Float32(2.0000000390829628e-24))
		tmp = Float32(s * Float32(Float32(4.0) * u));
	else
		tmp = Float32(Float32(Float32(s * s) * Float32(Float32(-1.0) / s)) / Float32(Float32(0.5) - Float32(Float32(0.25) / u)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((s * log((single(1.0) / (single(1.0) - (single(4.0) * u))))) <= single(2.0000000390829628e-24))
		tmp = s * (single(4.0) * u);
	else
		tmp = ((s * s) * (single(-1.0) / s)) / (single(0.5) - (single(0.25) / u));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 s (log.f32 (/.f32 #s(literal 1 binary32) (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))))) < 2.00000004e-24

   1. Initial program 58.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. lower-*.f32 79.4

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

   5. Applied rewrites 79.4%

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

   if 2.00000004e-24 < (*.f32 s (log.f32 (/.f32 #s(literal 1 binary32) (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)))))

   1. Initial program 66.3%

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

   4. Applied rewrites 44.8%

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

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

   6. Applied rewrites 66.7%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - s}}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}} \]

      3. metadata-eval N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\log 1 \cdot \log 1 - s \cdot s\right) \cdot \frac{1}{\color{blue}{0} + s}}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}} \]

      7. +-lft-identity N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{0} \cdot \log 1 - s \cdot s\right) \cdot \frac{1}{s}}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left(0 \cdot \color{blue}{0} - s \cdot s\right) \cdot \frac{1}{s}}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left(\color{blue}{0} - s \cdot s\right) \cdot \frac{1}{s}}{\frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}} \]

      12. sub0-neg N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-/.f32 66.6

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

   8. Applied rewrites 66.6%

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

   9. Taylor expanded in u around 0

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

       1. div-sub N/A

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

       2. associate-/l* N/A

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

       3. *-inverses N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. associate-*r/ N/A

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

       7. lower--.f32 N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f32 86.8

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

   11. Applied rewrites 86.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot \frac{-1}{s}}{0.5 - \frac{0.25}{u}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9998599886894226:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9998599886894226) (* s (log (/ 1.0 t_0))) (* s (* 4.0 u)))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9998599886894226f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * (4.0f * 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 - (4.0e0 * u)
    if (t_0 <= 0.9998599886894226e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = s * (4.0e0 * u)
    end if
    code = tmp
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9998599886894226))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32(4.0) * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	t_0 = single(1.0) - (single(4.0) * u);
	tmp = single(0.0);
	if (t_0 <= single(0.9998599886894226))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = s * (single(4.0) * 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)) < 0.999859989

   1. Initial program 84.3%

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

   if 0.999859989 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 45.0%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

Alternative 4: 74.0% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 73.6

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

5. Applied rewrites 73.6%

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

Reproduce

herbie shell --seed 2024307 
(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))))))