Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.8% → 97.5%

Time: 8.9s

Alternatives: 7

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: 60.8% 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: 97.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9819999933242798:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(\frac{\frac{4}{u} + 8}{u} + 21.333333333333332\right) \cdot {u}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9819999933242798)
     (* s (log (/ 1.0 t_0)))
     (* s (* (+ (/ (+ (/ 4.0 u) 8.0) u) 21.333333333333332) (pow u 3.0))))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9819999933242798f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * (((((4.0f / u) + 8.0f) / u) + 21.333333333333332f) * powf(u, 3.0f));
	}
	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 <= 0.9819999933242798e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = s * (((((4.0e0 / u) + 8.0e0) / u) + 21.333333333333332e0) * (u ** 3.0e0))
    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(0.9819999933242798))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32(Float32(Float32(Float32(Float32(4.0) / u) + Float32(8.0)) / u) + Float32(21.333333333333332)) * (u ^ Float32(3.0))));
	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(0.9819999933242798))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = s * (((((single(4.0) / u) + single(8.0)) / u) + single(21.333333333333332)) * (u ^ single(3.0)));
	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)) < 0.981999993

   1. Initial program 94.1%

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

   if 0.981999993 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 55.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 81.1

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

   10. Applied rewrites 80.7%

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

   11. Taylor expanded in u around inf

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

   13. Applied rewrites 98.1%

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

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t\_0 \leq 0.9819999933242798:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(\left(\frac{\frac{4}{u} + 8}{u} + 21.333333333333332\right) \cdot \left(u \cdot u\right)\right) \cdot u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9819999933242798)
     (* s (log (/ 1.0 t_0)))
     (* s (* (* (+ (/ (+ (/ 4.0 u) 8.0) u) 21.333333333333332) (* u u)) u)))))

C

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9819999933242798f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * ((((((4.0f / u) + 8.0f) / u) + 21.333333333333332f) * (u * u)) * 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 <= 0.9819999933242798e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = s * ((((((4.0e0 / u) + 8.0e0) / u) + 21.333333333333332e0) * (u * u)) * 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(0.9819999933242798))
		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(8.0)) / u) + Float32(21.333333333333332)) * Float32(u * u)) * 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(0.9819999933242798))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = s * ((((((single(4.0) / u) + single(8.0)) / u) + single(21.333333333333332)) * (u * u)) * 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)) < 0.981999993

   1. Initial program 94.1%

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

   if 0.981999993 < (-.f32 #s(literal 1 binary32) (*.f32 #s(literal 4 binary32) u))

   1. Initial program 55.2%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 81.1

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 81.1

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

   10. Applied rewrites 80.7%

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

   11. Taylor expanded in u around inf

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

   13. Applied rewrites 97.9%

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

Alternative 3: 90.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((((((4.0e0 / u) + 8.0e0) / u) + 21.333333333333332e0) * (u * u)) * u)
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * ((((((single(4.0) / u) + single(8.0)) / u) + single(21.333333333333332)) * (u * u)) * u);
end

Derivation

1. Initial program 63.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.f32 71.9

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

5. Applied rewrites 71.9%

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

7. Applied rewrites 71.8%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 71.9

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

10. Applied rewrites 71.9%

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

11. Taylor expanded in u around inf

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

13. Applied rewrites 90.0%

    \[\leadsto s \cdot \left(\left(\left(\frac{\frac{4}{u} + 8}{u} + 21.333333333333332\right) \cdot \left(u \cdot u\right)\right) \cdot u\right) \]
14. Add Preprocessing

Alternative 4: 85.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\left(-8 \cdot u - 4\right) \cdot u} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (* (- (* -64.0 u) 16.0) u) u) (/ s (* (- (* -8.0 u) 4.0) u))))

C

float code(float s, float u) {
	return ((((-64.0f * u) - 16.0f) * u) * u) * (s / (((-8.0f * u) - 4.0f) * u));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((((-64.0e0) * u) - 16.0e0) * u) * u) * (s / ((((-8.0e0) * u) - 4.0e0) * u))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(Float32(-64.0) * u) - Float32(16.0)) * u) * u) * Float32(s / Float32(Float32(Float32(Float32(-8.0) * u) - Float32(4.0)) * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = ((((single(-64.0) * u) - single(16.0)) * u) * u) * (s / (((single(-8.0) * u) - single(4.0)) * u));
end

Derivation

1. Initial program 63.7%

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

4. Applied rewrites 50.0%

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

5. Taylor expanded in u around 0

   \[\leadsto \color{blue}{\left({u}^{2} \cdot \left(-64 \cdot u - 16\right)\right)} \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot {u}^{2}\right)} \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(\left(-64 \cdot u - 16\right) \cdot \color{blue}{\left(u \cdot u\right)}\right) \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right)} \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right)} \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\left(-64 \cdot u - 16\right) \cdot u\right)} \cdot u\right) \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   6. lower--.f32 N/A

      \[\leadsto \left(\left(\color{blue}{\left(-64 \cdot u - 16\right)} \cdot u\right) \cdot u\right) \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

   7. lower-*.f32 72.4

      \[\leadsto \left(\left(\left(\color{blue}{-64 \cdot u} - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

7. Applied rewrites 72.2%

   \[\leadsto \color{blue}{\left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right)} \cdot \frac{s}{\mathsf{log1p}\left(-4 \cdot u\right)} \]

8. Taylor expanded in u around 0

   \[\leadsto \left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\color{blue}{u \cdot \left(-8 \cdot u - 4\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]

   3. lower--.f32 N/A

      \[\leadsto \left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\color{blue}{\left(-8 \cdot u - 4\right)} \cdot u} \]

   4. lower-*.f32 84.0

      \[\leadsto \left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\left(\color{blue}{-8 \cdot u} - 4\right) \cdot u} \]

10. Applied rewrites 84.0%

    \[\leadsto \left(\left(\left(-64 \cdot u - 16\right) \cdot u\right) \cdot u\right) \cdot \frac{s}{\color{blue}{\left(-8 \cdot u - 4\right) \cdot u}} \]
11. Add Preprocessing

Alternative 5: 47.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.f32 71.9

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

5. Applied rewrites 71.9%

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

7. Applied rewrites 71.8%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 71.9

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

10. Applied rewrites 71.9%

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

12. Applied rewrites 85.5%

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

Alternative 6: 47.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 63.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.f32 71.9

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

5. Applied rewrites 71.9%

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

7. Applied rewrites 71.8%

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

8. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 71.9

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

10. Applied rewrites 71.9%

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

12. Applied rewrites 85.3%

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

Alternative 7: 74.2% 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 63.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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. lower-*.f32 71.9

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

5. Applied rewrites 71.9%

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

Reproduce

herbie shell --seed 2024343 
(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))))))