Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.0% → 96.8%

Time: 6.6s

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: 61.0% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.00559999980032444:\\ \;\;\;\;\frac{s}{\frac{0.25}{u} - 0.5}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - e^{\log \left(u \cdot 4\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* 4.0 u) 0.00559999980032444)
   (/ s (- (/ 0.25 u) 0.5))
   (* s (log (/ 1.0 (- 1.0 (exp (log (* u 4.0)))))))))

C

float code(float s, float u) {
	float tmp;
	if ((4.0f * u) <= 0.00559999980032444f) {
		tmp = s / ((0.25f / u) - 0.5f);
	} else {
		tmp = s * logf((1.0f / (1.0f - expf(logf((u * 4.0f))))));
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((4.0e0 * u) <= 0.00559999980032444e0) then
        tmp = s / ((0.25e0 / u) - 0.5e0)
    else
        tmp = s * log((1.0e0 / (1.0e0 - exp(log((u * 4.0e0))))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(4.0) * u) <= Float32(0.00559999980032444))
		tmp = Float32(s / Float32(Float32(Float32(0.25) / u) - Float32(0.5)));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - exp(log(Float32(u * Float32(4.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(4.0) * u) <= single(0.00559999980032444))
		tmp = s / ((single(0.25) / u) - single(0.5));
	else
		tmp = s * log((single(1.0) / (single(1.0) - exp(log((u * single(4.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.0055999998

   1. Initial program 49.4%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 59.8%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{\frac{1}{4} + \frac{-1}{2} \cdot u}{u}}} \]
   6. 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 84.8

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 98.3%

       \[\leadsto s \cdot \frac{1}{\frac{0.25}{u} - \color{blue}{0.5}} \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. un-div-inv N/A

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

       4. lower-/.f32 98.4

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

   12. Applied rewrites 98.4%

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

   if 0.0055999998 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 94.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 94.2

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

      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 94.2

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

   4. Applied rewrites 94.2%

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

Alternative 2: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.00559999980032444:\\ \;\;\;\;\frac{s}{\frac{0.25}{u} - 0.5}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* 4.0 u) 0.00559999980032444)
   (/ s (- (/ 0.25 u) 0.5))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if ((4.0f * u) <= 0.00559999980032444f) {
		tmp = s / ((0.25f / u) - 0.5f);
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((4.0e0 * u) <= 0.00559999980032444e0) then
        tmp = s / ((0.25e0 / u) - 0.5e0)
    else
        tmp = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(4.0) * u) <= Float32(0.00559999980032444))
		tmp = Float32(s / Float32(Float32(Float32(0.25) / u) - Float32(0.5)));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(4.0) * u) <= single(0.00559999980032444))
		tmp = s / ((single(0.25) / u) - single(0.5));
	else
		tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.0055999998

   1. Initial program 49.4%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 58.6%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{\frac{1}{4} + \frac{-1}{2} \cdot u}{u}}} \]
   6. 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 84.8

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in u around inf

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

   10. Applied rewrites 98.3%

       \[\leadsto s \cdot \frac{1}{\frac{0.25}{u} - \color{blue}{0.5}} \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. un-div-inv N/A

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

       4. lower-/.f32 98.4

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

   12. Applied rewrites 98.4%

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

   if 0.0055999998 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 94.2%

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

Alternative 3: 88.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 47.1%

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

5. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{\frac{1}{4} + \frac{-1}{2} \cdot u}{u}}} \]
6. 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 73.7

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

7. Applied rewrites 73.7%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 88.0%

    \[\leadsto s \cdot \frac{1}{\frac{0.25}{u} - \color{blue}{0.5}} \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. un-div-inv N/A

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

    4. lower-/.f32 88.1

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

12. Applied rewrites 88.1%

    \[\leadsto \color{blue}{\frac{s}{\frac{0.25}{u} - 0.5}} \]
13. Add Preprocessing

Alternative 4: 73.9% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.3%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.f32 73.8

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

5. Applied rewrites 73.8%

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

Reproduce

herbie shell --seed 2024313 
(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))))))