Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 88.9%

Time: 6.3s

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: 61.3% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0001500000071246177:\\ \;\;\;\;\left(\frac{-0.25}{u} \cdot \left(-16 \cdot \left(u \cdot u\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.0001500000071246177)
   (* (* (/ -0.25 u) (* -16.0 (* u 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) <= 0.0001500000071246177f) {
		tmp = ((-0.25f / u) * (-16.0f * (u * 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) <= 0.0001500000071246177e0) then
        tmp = (((-0.25e0) / u) * ((-16.0e0) * (u * 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(0.0001500000071246177))
		tmp = Float32(Float32(Float32(Float32(-0.25) / u) * Float32(Float32(-16.0) * Float32(u * 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(0.0001500000071246177))
		tmp = ((single(-0.25) / u) * (single(-16.0) * (u * 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) < 1.50000007e-4

   1. Initial program 46.2%

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

      1. lift-log.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. log-div N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. lower-*.f32 N/A

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

   4. Applied rewrites 71.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f32 N/A

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

      5. inv-pow N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f32 71.8

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

   6. Applied rewrites 72.5%

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

   7. Taylor expanded in u around 0

      \[\leadsto s \cdot \left(\color{blue}{\left(-16 \cdot {u}^{2}\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      4. lower-*.f32 91.2

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   9. Applied rewrites 91.2%

      \[\leadsto s \cdot \left(\color{blue}{\left(\left(u \cdot u\right) \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   10. Taylor expanded in u around 0

       \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{\frac{-1}{4}}{u}}\right) \]
   11. Step-by-step derivation

       1. lower-/.f32 91.8

          \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{-0.25}{u}}\right) \]

   12. Applied rewrites 91.8%

       \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{-0.25}{u}}\right) \]

   if 1.50000007e-4 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 86.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 86.3

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

      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 86.3

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

   4. Applied rewrites 86.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0001500000071246177:\\ \;\;\;\;\left(\frac{-0.25}{u} \cdot \left(-16 \cdot \left(u \cdot u\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0001500000071246177:\\ \;\;\;\;\left(\frac{-0.25}{u} \cdot \left(-16 \cdot \left(u \cdot u\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.0001500000071246177)
   (* (* (/ -0.25 u) (* -16.0 (* u u))) s)
   (* (log (/ 1.0 (- 1.0 (* u 4.0)))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.0001500000071246177f) {
		tmp = ((-0.25f / u) * (-16.0f * (u * u))) * s;
	} else {
		tmp = logf((1.0f / (1.0f - (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) <= 0.0001500000071246177e0) then
        tmp = (((-0.25e0) / u) * ((-16.0e0) * (u * u))) * s
    else
        tmp = log((1.0e0 / (1.0e0 - (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(0.0001500000071246177))
		tmp = Float32(Float32(Float32(Float32(-0.25) / u) * Float32(Float32(-16.0) * Float32(u * u))) * s);
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - 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(0.0001500000071246177))
		tmp = ((single(-0.25) / u) * (single(-16.0) * (u * u))) * s;
	else
		tmp = log((single(1.0) / (single(1.0) - (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) < 1.50000007e-4

   1. Initial program 46.2%

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

      1. lift-log.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. log-div N/A

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

      4. flip-- N/A

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

      5. div-inv N/A

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

      6. lower-*.f32 N/A

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

   4. Applied rewrites 71.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f32 N/A

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

      5. inv-pow N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f32 71.4

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

   6. Applied rewrites 72.0%

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

   7. Taylor expanded in u around 0

      \[\leadsto s \cdot \left(\color{blue}{\left(-16 \cdot {u}^{2}\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      4. lower-*.f32 91.2

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   9. Applied rewrites 91.2%

      \[\leadsto s \cdot \left(\color{blue}{\left(\left(u \cdot u\right) \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   10. Taylor expanded in u around 0

       \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{\frac{-1}{4}}{u}}\right) \]
   11. Step-by-step derivation

       1. lower-/.f32 91.8

          \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{-0.25}{u}}\right) \]

   12. Applied rewrites 91.8%

       \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{-0.25}{u}}\right) \]

   if 1.50000007e-4 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 86.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0001500000071246177:\\ \;\;\;\;\left(\frac{-0.25}{u} \cdot \left(-16 \cdot \left(u \cdot u\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{-0.25}{u} \cdot \left(-16 \cdot \left(u \cdot u\right)\right)\right) \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (/ -0.25 u) (* -16.0 (* u u))) s))

C

float code(float s, float u) {
	return ((-0.25f / u) * (-16.0f * (u * u))) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((-0.25e0) / u) * ((-16.0e0) * (u * u))) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(-0.25) / u) * Float32(Float32(-16.0) * Float32(u * u))) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = ((single(-0.25) / u) * (single(-16.0) * (u * u))) * s;
end

Derivation

1. Initial program 63.1%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. log-div N/A

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

   4. flip-- N/A

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

   5. div-inv N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 45.9%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lift-pow.f32 N/A

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

   5. inv-pow N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f32 45.1

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

6. Applied rewrites 46.8%

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

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(\color{blue}{\left(-16 \cdot {u}^{2}\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   4. lower-*.f32 73.4

      \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

9. Applied rewrites 73.4%

   \[\leadsto s \cdot \left(\color{blue}{\left(\left(u \cdot u\right) \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

10. Taylor expanded in u around 0

    \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{\frac{-1}{4}}{u}}\right) \]
11. Step-by-step derivation

    1. lower-/.f32 73.6

       \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{-0.25}{u}}\right) \]

12. Applied rewrites 73.6%

    \[\leadsto s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \color{blue}{\frac{-0.25}{u}}\right) \]

13. Final simplification 73.6%

    \[\leadsto \left(\frac{-0.25}{u} \cdot \left(-16 \cdot \left(u \cdot u\right)\right)\right) \cdot s \]
14. Add Preprocessing

Alternative 4: 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 63.1%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. 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. Final simplification 73.6%

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

Reproduce

herbie shell --seed 2024304 
(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))))))