Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 89.2%

Time: 7.4s

Alternatives: 6

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: 61.1% 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: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9998000264167786:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9998000264167786) (* s (log (/ 1.0 t_0))) (* s (* 4.0 u)))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9998000264167786f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * (4.0f * u);
	}
	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 - (4.0e0 * u)
    if (t_0 <= 0.9998000264167786e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = s * (4.0e0 * u)
    end if
    code = tmp
end function

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9998000264167786))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32(4.0) * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	t_0 = single(1.0) - (single(4.0) * u);
	tmp = single(0.0);
	if (t_0 <= single(0.9998000264167786))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = s * (single(4.0) * u);
	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)) < 0.999800026

   1. Initial program 87.4%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 41.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 2: 58.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ t_1 := \mathsf{log1p}\left(-4 \cdot u\right)\\ \mathbf{if}\;t\_0 \leq 0.999779999256134:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left({\left(\left(\left(16 \cdot u\right) \cdot u\right) \cdot \frac{1}{t\_1}\right)}^{2} \cdot \frac{-1}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))) (t_1 (log1p (* -4.0 u))))
   (if (<= t_0 0.999779999256134)
     (* s (log (/ 1.0 t_0)))
     (* s (* (pow (* (* (* 16.0 u) u) (/ 1.0 t_1)) 2.0) (/ -1.0 t_1))))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float t_1 = log1pf((-4.0f * u));
	float tmp;
	if (t_0 <= 0.999779999256134f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * (powf((((16.0f * u) * u) * (1.0f / t_1)), 2.0f) * (-1.0f / t_1));
	}
	return tmp;
}

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	t_1 = log1p(Float32(Float32(-4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.999779999256134))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32((Float32(Float32(Float32(Float32(16.0) * u) * u) * Float32(Float32(1.0) / t_1)) ^ Float32(2.0)) * Float32(Float32(-1.0) / t_1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.999779999

   1. Initial program 87.8%

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

   if 0.999779999 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 42.2%

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

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

      6. lower-*.f32 N/A

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

   4. Applied rewrites 68.4%

      \[\leadsto s \cdot \color{blue}{\left(\left(-{\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)} \]
   5. Step-by-step derivation

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lift-pow.f32 N/A

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

      5. inv-pow N/A

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

      6. lift-/.f32 N/A

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

      7. lower-*.f32 68.3

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

   6. Applied rewrites 69.6%

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

   7. Taylor expanded in u around 0

      \[\leadsto s \cdot \left(\left(-{\left(\color{blue}{\left(16 \cdot {u}^{2}\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto s \cdot \left(\left(-{\left(\left(16 \cdot \color{blue}{\left(u \cdot u\right)}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto s \cdot \left(\left(-{\left(\color{blue}{\left(\left(16 \cdot u\right) \cdot u\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto s \cdot \left(\left(-{\left(\color{blue}{\left(\left(16 \cdot u\right) \cdot u\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      4. lower-*.f32 59.1

         \[\leadsto s \cdot \left(\left(-{\left(\left(\color{blue}{\left(16 \cdot u\right)} \cdot u\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   9. Applied rewrites 59.6%

      \[\leadsto s \cdot \left(\left(-{\left(\color{blue}{\left(\left(16 \cdot u\right) \cdot u\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.999779999256134:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left({\left(\left(\left(16 \cdot u\right) \cdot u\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)}^{2} \cdot \frac{-1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9997199773788452:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left({u}^{3} \cdot 64\right) \cdot {\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9997199773788452)
     (* s (log (/ 1.0 t_0)))
     (* s (* (* (pow u 3.0) 64.0) (pow (log1p (* -4.0 u)) -2.0))))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9997199773788452f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * ((powf(u, 3.0f) * 64.0f) * powf(log1pf((-4.0f * u)), -2.0f));
	}
	return tmp;
}

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9997199773788452))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32((u ^ Float32(3.0)) * Float32(64.0)) * (log1p(Float32(Float32(-4.0) * u)) ^ Float32(-2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.999719977

   1. Initial program 88.2%

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

   if 0.999719977 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 42.8%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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. flip3-- N/A

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

      5. div-inv N/A

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

      6. lower-*.f32 N/A

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \left(\color{blue}{\left(64 \cdot {u}^{3}\right)} \cdot {\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{-2}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{3} \cdot 64\right)} \cdot {\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{-2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{3} \cdot 64\right)} \cdot {\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{-2}\right) \]

      3. lower-pow.f32 90.6

         \[\leadsto s \cdot \left(\left(\color{blue}{{u}^{3}} \cdot 64\right) \cdot {\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{-2}\right) \]

   7. Applied rewrites 90.6%

      \[\leadsto s \cdot \left(\color{blue}{\left({u}^{3} \cdot 64\right)} \cdot {\left(\mathsf{log1p}\left(-4 \cdot u\right)\right)}^{-2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9997000098228455:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(\left(u \cdot u\right) \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9997000098228455)
     (* s (log (/ 1.0 t_0)))
     (* s (* (* (* u u) -16.0) (/ 1.0 (log1p (* -4.0 u))))))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9997000098228455f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * (((u * u) * -16.0f) * (1.0f / log1pf((-4.0f * u))));
	}
	return tmp;
}

Julia

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9997000098228455))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32(Float32(u * u) * Float32(-16.0)) * Float32(Float32(1.0) / log1p(Float32(Float32(-4.0) * u)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u)) < 0.99970001

   1. Initial program 88.4%

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

   if 0.99970001 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 42.9%

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

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

      6. lower-*.f32 N/A

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \left(\color{blue}{\left(-16 \cdot {u}^{2}\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\left({u}^{2} \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

      4. lower-*.f32 89.5

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -16\right) \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]

   7. Applied rewrites 89.5%

      \[\leadsto s \cdot \left(\color{blue}{\left(\left(u \cdot u\right) \cdot -16\right)} \cdot \frac{1}{\mathsf{log1p}\left(-4 \cdot u\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9998000264167786:\\ \;\;\;\;s \cdot \log \left(\left(16 \cdot u + 4\right) \cdot u + 1\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (- 1.0 (* 4.0 u)) 0.9998000264167786)
   (* s (log (+ (* (+ (* 16.0 u) 4.0) u) 1.0)))
   (* s (* 4.0 u))))

C

float code(float s, float u) {
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9998000264167786f) {
		tmp = s * logf(((((16.0f * u) + 4.0f) * u) + 1.0f));
	} else {
		tmp = s * (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 ((1.0e0 - (4.0e0 * u)) <= 0.9998000264167786e0) then
        tmp = s * log(((((16.0e0 * u) + 4.0e0) * u) + 1.0e0))
    else
        tmp = s * (4.0e0 * u)
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9998000264167786))
		tmp = Float32(s * log(Float32(Float32(Float32(Float32(Float32(16.0) * u) + Float32(4.0)) * u) + Float32(1.0))));
	else
		tmp = Float32(s * Float32(Float32(4.0) * u));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((single(1.0) - (single(4.0) * u)) <= single(0.9998000264167786))
		tmp = s * log(((((single(16.0) * u) + single(4.0)) * u) + single(1.0)));
	else
		tmp = s * (single(4.0) * u);
	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)) < 0.999800026

   1. Initial program 87.4%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f32 N/A

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

      8. metadata-eval 12.8

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

   4. Applied rewrites 12.8%

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

   5. Taylor expanded in u around 0

      \[\leadsto s \cdot \log \color{blue}{\left(1 + u \cdot \left(4 + 16 \cdot u\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto s \cdot \log \color{blue}{\left(u \cdot \left(4 + 16 \cdot u\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto s \cdot \log \left(\color{blue}{\left(4 + 16 \cdot u\right) \cdot u} + 1\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto s \cdot \log \color{blue}{\left(\mathsf{fma}\left(4 + 16 \cdot u, u, 1\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto s \cdot \log \left(\mathsf{fma}\left(\color{blue}{16 \cdot u + 4}, u, 1\right)\right) \]

      5. lower-fma.f32 12.8

         \[\leadsto s \cdot \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(16, u, 4\right)}, u, 1\right)\right) \]

   7. Applied rewrites 12.8%

      \[\leadsto s \cdot \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(16, u, 4\right), u, 1\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.7%

      \[\leadsto s \cdot \log \left(\mathsf{fma}\left(16, u, 4\right) \cdot u + \color{blue}{1}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 58.5%

       \[\leadsto s \cdot \log \left(\left(16 \cdot u + 4\right) \cdot u + 1\right) \]

   if 0.999800026 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 41.5%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 6: 74.0% 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 61.2%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.8

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

5. Applied rewrites 72.8%

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

Reproduce

herbie shell --seed 2024322 
(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))))))