Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.9% → 96.8%

Time: 6.4s

Alternatives: 3

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: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0038399999029934406:\\ \;\;\;\;\frac{1}{\frac{0.25}{u} - 0.5} \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.0038399999029934406)
   (* (/ 1.0 (- (/ 0.25 u) 0.5)) s)
   (* (log (/ 1.0 (- 1.0 (* u 4.0)))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.0038399999029934406f) {
		tmp = (1.0f / ((0.25f / u) - 0.5f)) * 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) <= 0.0038399999029934406e0) then
        tmp = (1.0e0 / ((0.25e0 / u) - 0.5e0)) * 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(0.0038399999029934406))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(Float32(0.25) / u) - Float32(0.5))) * 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(0.0038399999029934406))
		tmp = (single(1.0) / ((single(0.25) / u) - single(0.5))) * 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) < 0.0038399999

   1. Initial program 50.3%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 61.3%

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

   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 86.3

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in u around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 45.0

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

   10. Applied rewrites 85.9%

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

   11. Taylor expanded in u around inf

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

   13. Applied rewrites 98.7%

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

   if 0.0038399999 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 92.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.0038399999029934406:\\ \;\;\;\;\frac{1}{\frac{0.25}{u} - 0.5} \cdot s\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 46.2%

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

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 74.0

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

7. Applied rewrites 74.0%

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

8. Taylor expanded in u around 0

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

   1. lower-/.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 74.0

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

10. Applied rewrites 73.7%

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

11. Taylor expanded in u around inf

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

13. Applied rewrites 88.4%

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

14. Final simplification 88.4%

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

Alternative 3: 73.9% 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 62.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 74.0

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

5. Applied rewrites 74.0%

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

6. Final simplification 74.0%

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

Reproduce

herbie shell --seed 2024264 
(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))))))