Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.9% → 92.8%

Time: 7.5s

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: 61.9% 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: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.9959999918937683:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;e^{2 \cdot u - \left(\log 0.25 - \log u\right)} \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.9959999918937683)
     (* (log (/ 1.0 t_0)) s)
     (* (exp (- (* 2.0 u) (- (log 0.25) (log u)))) s))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (u * 4.0f);
	float tmp;
	if (t_0 <= 0.9959999918937683f) {
		tmp = logf((1.0f / t_0)) * s;
	} else {
		tmp = expf(((2.0f * u) - (logf(0.25f) - logf(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 <= 0.9959999918937683e0) then
        tmp = log((1.0e0 / t_0)) * s
    else
        tmp = exp(((2.0e0 * u) - (log(0.25e0) - log(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(0.9959999918937683))
		tmp = Float32(log(Float32(Float32(1.0) / t_0)) * s);
	else
		tmp = Float32(exp(Float32(Float32(Float32(2.0) * u) - Float32(log(Float32(0.25)) - log(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(0.9959999918937683))
		tmp = log((single(1.0) / t_0)) * s;
	else
		tmp = exp(((single(2.0) * u) - (log(single(0.25)) - log(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)) < 0.995999992

   1. Initial program 92.3%

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

   if 0.995999992 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 49.4%

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in u around 0

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

      1. lower-/.f32 85.1

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

   7. Applied rewrites 85.1%

      \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{0.25}{u}}} \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-log.f32 83.2

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

   9. Applied rewrites 83.2%

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

   10. Taylor expanded in u around 0

       \[\leadsto s \cdot e^{\color{blue}{-1 \cdot \left(\log \frac{1}{4} + -1 \cdot \log u\right) + 2 \cdot u}} \]
   11. 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 92.8

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

   12. Applied rewrites 92.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9959999918937683:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;e^{2 \cdot u - \left(\log 0.25 - \log u\right)} \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.999750018119812:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.25}{u}} \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.999750018119812)
     (* (log (/ 1.0 t_0)) s)
     (* (/ 1.0 (/ 0.25 u)) s))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (u * 4.0f);
	float tmp;
	if (t_0 <= 0.999750018119812f) {
		tmp = logf((1.0f / t_0)) * s;
	} else {
		tmp = (1.0f / (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 <= 0.999750018119812e0) then
        tmp = log((1.0e0 / t_0)) * s
    else
        tmp = (1.0e0 / (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(0.999750018119812))
		tmp = Float32(log(Float32(Float32(1.0) / t_0)) * s);
	else
		tmp = Float32(Float32(Float32(1.0) / 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(0.999750018119812))
		tmp = log((single(1.0) / t_0)) * s;
	else
		tmp = (single(1.0) / (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)) < 0.999750018

   1. Initial program 87.2%

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

   if 0.999750018 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 42.3%

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in u around 0

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

      1. lower-/.f32 91.0

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

   7. Applied rewrites 91.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.999750018119812:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.25}{u}} \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.0%

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in u around 0

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

   1. lower-/.f32 72.1

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

7. Applied rewrites 72.1%

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

8. Final simplification 72.1%

   \[\leadsto \frac{1}{\frac{0.25}{u}} \cdot s \]
9. Add Preprocessing

Alternative 4: 73.6% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.0%

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

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

   2. lower-*.f32 72.1

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

5. Applied rewrites 72.1%

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

6. Final simplification 72.1%

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

Reproduce

herbie shell --seed 2024266 
(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))))))