Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 98.2%

Time: 6.7s

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: 61.4% 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: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9599999785423279:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(\left(\left(\left(\frac{4}{{u}^{3}} + \frac{8}{u \cdot u}\right) + \frac{21.333333333333332}{u}\right) + 64\right) \cdot {u}^{3}\right) \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9599999785423279)
     (* s (log (/ 1.0 t_0)))
     (*
      s
      (*
       (*
        (+
         (+ (+ (/ 4.0 (pow u 3.0)) (/ 8.0 (* u u))) (/ 21.333333333333332 u))
         64.0)
        (pow u 3.0))
       u)))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9599999785423279f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * ((((((4.0f / powf(u, 3.0f)) + (8.0f / (u * u))) + (21.333333333333332f / u)) + 64.0f) * powf(u, 3.0f)) * 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 <= 0.9599999785423279e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = s * ((((((4.0e0 / (u ** 3.0e0)) + (8.0e0 / (u * u))) + (21.333333333333332e0 / u)) + 64.0e0) * (u ** 3.0e0)) * 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(0.9599999785423279))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(Float32(4.0) / (u ^ Float32(3.0))) + Float32(Float32(8.0) / Float32(u * u))) + Float32(Float32(21.333333333333332) / u)) + Float32(64.0)) * (u ^ Float32(3.0))) * 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(0.9599999785423279))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = s * ((((((single(4.0) / (u ^ single(3.0))) + (single(8.0) / (u * u))) + (single(21.333333333333332) / u)) + single(64.0)) * (u ^ single(3.0))) * 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)) < 0.959999979

   1. Initial program 96.3%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 53.6%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

      6. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

      7. *-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

      9. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

      10. lower-fma.f32 81.2

          \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

   8. Applied rewrites 80.7%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]

   9. Taylor expanded in u around inf

      \[\leadsto s \cdot \left(\left({u}^{3} \cdot \left(64 + \left(\frac{64}{3} \cdot \frac{1}{u} + \left(4 \cdot \frac{1}{{u}^{3}} + \frac{8}{{u}^{2}}\right)\right)\right)\right) \cdot u\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto s \cdot \left(\left(\left(\left(\left(\frac{4}{{u}^{3}} + \frac{8}{u \cdot u}\right) + \frac{21.333333333333332}{u}\right) + 64\right) \cdot {u}^{3}\right) \cdot u\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9599999785423279:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(64 - \frac{-21.333333333333332 - \frac{\frac{4}{u} + 8}{u}}{u}\right) \cdot {u}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9599999785423279)
     (* s (log (/ 1.0 t_0)))
     (*
      s
      (*
       (- 64.0 (/ (- -21.333333333333332 (/ (+ (/ 4.0 u) 8.0) u)) u))
       (pow u 4.0))))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9599999785423279f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * ((64.0f - ((-21.333333333333332f - (((4.0f / u) + 8.0f) / u)) / u)) * powf(u, 4.0f));
	}
	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 <= 0.9599999785423279e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = s * ((64.0e0 - (((-21.333333333333332e0) - (((4.0e0 / u) + 8.0e0) / u)) / u)) * (u ** 4.0e0))
    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(0.9599999785423279))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32(Float32(64.0) - Float32(Float32(Float32(-21.333333333333332) - Float32(Float32(Float32(Float32(4.0) / u) + Float32(8.0)) / u)) / u)) * (u ^ Float32(4.0))));
	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(0.9599999785423279))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = s * ((single(64.0) - ((single(-21.333333333333332) - (((single(4.0) / u) + single(8.0)) / u)) / u)) * (u ^ single(4.0)));
	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)) < 0.959999979

   1. Initial program 96.3%

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

   if 0.959999979 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 53.6%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

      6. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

      7. *-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

      9. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

      10. lower-fma.f32 81.2

          \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

   8. Applied rewrites 80.7%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]

   9. Taylor expanded in u around inf

      \[\leadsto s \cdot \left(\left({u}^{3} \cdot \left(64 + \left(\frac{64}{3} \cdot \frac{1}{u} + \left(4 \cdot \frac{1}{{u}^{3}} + \frac{8}{{u}^{2}}\right)\right)\right)\right) \cdot u\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto s \cdot \left(\left(\left(\left(\left(\frac{4}{{u}^{3}} + \frac{8}{u \cdot u}\right) + \frac{21.333333333333332}{u}\right) + 64\right) \cdot {u}^{3}\right) \cdot u\right) \]

   12. Taylor expanded in u around -inf

       \[\leadsto s \cdot \left({u}^{4} \cdot \color{blue}{\left(64 + -1 \cdot \frac{-1 \cdot \frac{8 + 4 \cdot \frac{1}{u}}{u} - \frac{64}{3}}{u}\right)}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 98.7%

       \[\leadsto s \cdot \left(\left(64 - \frac{-21.333333333333332 - \frac{\frac{4}{u} + 8}{u}}{u}\right) \cdot \color{blue}{{u}^{4}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.009999999776482582:\\ \;\;\;\;s \cdot \left(\left(\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot u + 8\right) \cdot u\right) \cdot u + u \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* 4.0 u) 0.009999999776482582)
   (*
    s
    (+ (* (* (+ (* (fma 64.0 u 21.333333333333332) u) 8.0) u) u) (* u 4.0)))
   (* s (log (/ 1.0 (- 1.0 (* 4.0 u)))))))

C

float code(float s, float u) {
	float tmp;
	if ((4.0f * u) <= 0.009999999776482582f) {
		tmp = s * (((((fmaf(64.0f, u, 21.333333333333332f) * u) + 8.0f) * u) * u) + (u * 4.0f));
	} else {
		tmp = s * logf((1.0f / (1.0f - (4.0f * u))));
	}
	return tmp;
}

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(Float32(4.0) * u) <= Float32(0.009999999776482582))
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(fma(Float32(64.0), u, Float32(21.333333333333332)) * u) + Float32(8.0)) * u) * u) + Float32(u * Float32(4.0))));
	else
		tmp = Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.00999999978

   1. Initial program 50.6%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 84.1

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in u around 0

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

      3. +-commutative N/A

         \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

      4. *-commutative N/A

         \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

      6. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

      7. *-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

      9. +-commutative N/A

         \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

      10. lower-fma.f32 84.1

          \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

   8. Applied rewrites 83.6%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.3%

       \[\leadsto s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u\right) \cdot u + \color{blue}{u \cdot 4}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 98.5%

       \[\leadsto s \cdot \left(\left(\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot u + 8\right) \cdot u\right) \cdot u + u \cdot 4\right) \]

   if 0.00999999978 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 93.1%

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

Alternative 4: 58.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot u + 8\right) \cdot u\right) \cdot u + u \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (+ (* (* (+ (* (fma 64.0 u 21.333333333333332) u) 8.0) u) u) (* u 4.0))))

C

float code(float s, float u) {
	return s * (((((fmaf(64.0f, u, 21.333333333333332f) * u) + 8.0f) * u) * u) + (u * 4.0f));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(Float32(Float32(fma(Float32(64.0), u, Float32(21.333333333333332)) * u) + Float32(8.0)) * u) * u) + Float32(u * Float32(4.0))))
end

Derivation

1. Initial program 60.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.9

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

5. Applied rewrites 73.9%

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

6. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   3. +-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

   4. *-commutative N/A

      \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

   6. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

   7. *-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

   9. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. lower-fma.f32 73.9

       \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

8. Applied rewrites 73.9%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
9. Step-by-step derivation

10. Applied rewrites 86.8%

    \[\leadsto s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u\right) \cdot u + \color{blue}{u \cdot 4}\right) \]
11. Step-by-step derivation

12. Applied rewrites 90.3%

    \[\leadsto s \cdot \left(\left(\left(\mathsf{fma}\left(64, u, 21.333333333333332\right) \cdot u + 8\right) \cdot u\right) \cdot u + u \cdot 4\right) \]
13. Add Preprocessing

Alternative 5: 86.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(8 \cdot u\right) \cdot u + u \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (+ (* (* 8.0 u) u) (* u 4.0))))

C

float code(float s, float u) {
	return s * (((8.0f * u) * u) + (u * 4.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (((8.0e0 * u) * u) + (u * 4.0e0))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(Float32(8.0) * u) * u) + Float32(u * Float32(4.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (((single(8.0) * u) * u) + (u * single(4.0)));
end

Derivation

1. Initial program 60.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.9

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

5. Applied rewrites 73.9%

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

6. Taylor expanded in u around 0

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right) \cdot u\right)} \]

   3. +-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)} \cdot u\right) \]

   4. *-commutative N/A

      \[\leadsto s \cdot \left(\left(\color{blue}{\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) \cdot u} + 4\right) \cdot u\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), u, 4\right)} \cdot u\right) \]

   6. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, u, 4\right) \cdot u\right) \]

   7. *-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{64}{3} + 64 \cdot u\right) \cdot u} + 8, u, 4\right) \cdot u\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64}{3} + 64 \cdot u, u, 8\right)}, u, 4\right) \cdot u\right) \]

   9. +-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{64 \cdot u + \frac{64}{3}}, u, 8\right), u, 4\right) \cdot u\right) \]

   10. lower-fma.f32 73.9

       \[\leadsto s \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(64, u, 21.333333333333332\right)}, u, 8\right), u, 4\right) \cdot u\right) \]

8. Applied rewrites 73.9%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right), u, 4\right) \cdot u\right)} \]
9. Step-by-step derivation

10. Applied rewrites 86.8%

    \[\leadsto s \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(64, u, 21.333333333333332\right), u, 8\right) \cdot u\right) \cdot u + \color{blue}{u \cdot 4}\right) \]

11. Taylor expanded in u around 0

    \[\leadsto s \cdot \left(\left(8 \cdot u\right) \cdot u + u \cdot 4\right) \]
12. Step-by-step derivation

13. Applied rewrites 86.8%

    \[\leadsto s \cdot \left(\left(8 \cdot u\right) \cdot u + u \cdot 4\right) \]
14. Add Preprocessing

Alternative 6: 73.8% 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 60.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 73.9

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

5. Applied rewrites 73.9%

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

Reproduce

herbie shell --seed 2024298 
(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))))))