Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 93.3%

Time: 6.2s

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: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.006000000052154064:\\ \;\;\;\;e^{2 \cdot u - \log \left(\frac{0.25}{u}\right)} \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.006000000052154064)
   (* (exp (- (* 2.0 u) (log (/ 0.25 u)))) s)
   (* (log (/ 1.0 (- 1.0 (* u 4.0)))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.006000000052154064f) {
		tmp = expf(((2.0f * u) - logf((0.25f / 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) <= 0.006000000052154064e0) then
        tmp = exp(((2.0e0 * u) - log((0.25e0 / 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(0.006000000052154064))
		tmp = Float32(exp(Float32(Float32(Float32(2.0) * u) - log(Float32(Float32(0.25) / 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(0.006000000052154064))
		tmp = exp(((single(2.0) * u) - log((single(0.25) / 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) < 0.00600000005

   1. Initial program 49.8%

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

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

      6. lower-/.f32 N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 60.4%

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

   6. Applied rewrites 9.8%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. lower--.f32 N/A

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

      14. lower-log.f32 N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

      20. lower-log.f32 93.3

          \[\leadsto s \cdot e^{2 \cdot u - \left(\log 0.25 - \color{blue}{\log u}\right)} \]

   9. Applied rewrites 93.3%

      \[\leadsto s \cdot e^{\color{blue}{2 \cdot u - \left(\log 0.25 - \log u\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.1%

       \[\leadsto s \cdot e^{u \cdot 2 - \color{blue}{\log \left(\frac{0.25}{u}\right)}} \]

   if 0.00600000005 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 92.6%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.006000000052154064:\\ \;\;\;\;e^{2 \cdot u - \log \left(\frac{0.25}{u}\right)} \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.9940000176429749:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(u \cdot 4\right) + 2 \cdot u} \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.9940000176429749)
     (* (log (/ 1.0 t_0)) s)
     (* (exp (+ (log (* u 4.0)) (* 2.0 u))) s))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (u * 4.0f);
	float tmp;
	if (t_0 <= 0.9940000176429749f) {
		tmp = logf((1.0f / t_0)) * s;
	} else {
		tmp = expf((logf((u * 4.0f)) + (2.0f * 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 <= 0.9940000176429749e0) then
        tmp = log((1.0e0 / t_0)) * s
    else
        tmp = exp((log((u * 4.0e0)) + (2.0e0 * 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(0.9940000176429749))
		tmp = Float32(log(Float32(Float32(1.0) / t_0)) * s);
	else
		tmp = Float32(exp(Float32(log(Float32(u * Float32(4.0))) + Float32(Float32(2.0) * 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(0.9940000176429749))
		tmp = log((single(1.0) / t_0)) * s;
	else
		tmp = exp((log((u * single(4.0))) + (single(2.0) * 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)) < 0.994000018

   1. Initial program 92.6%

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

   if 0.994000018 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 49.8%

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

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

      6. lower-/.f32 N/A

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

      7. metadata-eval N/A

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

      8. +-lft-identity N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 60.4%

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

   6. Applied rewrites 11.0%

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

   7. Taylor expanded in u around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. lower--.f32 N/A

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

      14. lower-log.f32 N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. mul-1-neg N/A

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

      19. remove-double-neg N/A

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

      20. lower-log.f32 93.3

          \[\leadsto s \cdot e^{2 \cdot u - \left(\log 0.25 - \color{blue}{\log u}\right)} \]

   9. Applied rewrites 93.3%

      \[\leadsto s \cdot e^{\color{blue}{2 \cdot u - \left(\log 0.25 - \log u\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9940000176429749:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(u \cdot 4\right) + 2 \cdot u} \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.9998660087585449:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.9998660087585449) (* (log (/ 1.0 t_0)) s) (* (* u 4.0) s))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (u * 4.0f);
	float tmp;
	if (t_0 <= 0.9998660087585449f) {
		tmp = logf((1.0f / t_0)) * s;
	} else {
		tmp = (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) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 - (u * 4.0e0)
    if (t_0 <= 0.9998660087585449e0) then
        tmp = log((1.0e0 / t_0)) * s
    else
        tmp = (u * 4.0e0) * 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(0.9998660087585449))
		tmp = Float32(log(Float32(Float32(1.0) / t_0)) * s);
	else
		tmp = Float32(Float32(u * Float32(4.0)) * 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(0.9998660087585449))
		tmp = log((single(1.0) / t_0)) * s;
	else
		tmp = (u * single(4.0)) * 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)) < 0.999866009

   1. Initial program 85.4%

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

   if 0.999866009 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 42.0%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9998660087585449:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% 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 60.5%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 74.1

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

5. Applied rewrites 74.1%

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

6. Final simplification 74.1%

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

Reproduce

herbie shell --seed 2024295 
(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))))))