Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 89.1%

Time: 6.9s

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: 61.2% 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: 89.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9997919797897339:\\ \;\;\;\;\log \left(\frac{1}{1 - \frac{1}{\frac{0.25}{u}}}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* u 4.0)) 0.9997919797897339)
   (* (log (/ 1.0 (- 1.0 (/ 1.0 (/ 0.25 u))))) s)
   (* (* u 4.0) s)))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (u * 4.0f)) <= 0.9997919797897339f) {
		tmp = logf((1.0f / (1.0f - (1.0f / (0.25f / u))))) * 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) :: tmp
    if ((1.0e0 - (u * 4.0e0)) <= 0.9997919797897339e0) then
        tmp = log((1.0e0 / (1.0e0 - (1.0e0 / (0.25e0 / u))))) * s
    else
        tmp = (u * 4.0e0) * s
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(u * Float32(4.0))) <= Float32(0.9997919797897339))
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(1.0) / Float32(Float32(0.25) / u))))) * s);
	else
		tmp = Float32(Float32(u * Float32(4.0)) * s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(1.0) - (u * single(4.0))) <= single(0.9997919797897339))
		tmp = log((single(1.0) / (single(1.0) - (single(1.0) / (single(0.25) / u))))) * 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)) < 0.99979198

   1. Initial program 86.9%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f32 N/A

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

      10. metadata-eval 86.9

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

   4. Applied rewrites 86.9%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval 86.9

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

   6. Applied rewrites 86.9%

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

   if 0.99979198 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 42.1%

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

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

      2. lower-*.f32 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9997919797897339:\\ \;\;\;\;\log \left(\frac{1}{1 - \frac{1}{\frac{0.25}{u}}}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u \cdot 4\\ \mathbf{if}\;t\_0 \leq 0.9997919797897339:\\ \;\;\;\;\log \left(\frac{1}{t\_0}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u 4.0))))
   (if (<= t_0 0.9997919797897339) (* (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 <= 0.9997919797897339f) {
		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 <= 0.9997919797897339e0) 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(0.9997919797897339))
		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(0.9997919797897339))
		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)) < 0.99979198

   1. Initial program 86.9%

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

   if 0.99979198 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 42.1%

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

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

      2. lower-*.f32 90.7

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

   5. Applied rewrites 90.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \cdot 4 \leq 0.9997919797897339:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(u \cdot 4\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% 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 59.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.8

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

5. Applied rewrites 73.8%

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

6. Final simplification 73.8%

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

Reproduce

herbie shell --seed 2024271 
(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))))))