Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 98.7%

Time: 8.7s

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: 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: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.03999999910593033:\\ \;\;\;\;\left(s \cdot 4 + \left(\left(\left(u \cdot u\right) \cdot \left(\frac{\frac{8}{u} + 21.333333333333332}{u} + 64\right)\right) \cdot u\right) \cdot s\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.03999999910593033)
   (*
    (+
     (* s 4.0)
     (* (* (* (* u u) (+ (/ (+ (/ 8.0 u) 21.333333333333332) u) 64.0)) u) s))
    u)
   (* (log (/ 1.0 (- 1.0 (* u 4.0)))) s)))

C

float code(float s, float u) {
	float tmp;
	if ((u * 4.0f) <= 0.03999999910593033f) {
		tmp = ((s * 4.0f) + ((((u * u) * ((((8.0f / u) + 21.333333333333332f) / u) + 64.0f)) * u) * s)) * u;
	} else {
		tmp = logf((1.0f / (1.0f - (u * 4.0f)))) * s;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if ((u * 4.0e0) <= 0.03999999910593033e0) then
        tmp = ((s * 4.0e0) + ((((u * u) * ((((8.0e0 / u) + 21.333333333333332e0) / u) + 64.0e0)) * u) * s)) * u
    else
        tmp = log((1.0e0 / (1.0e0 - (u * 4.0e0)))) * s
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (Float32(u * Float32(4.0)) <= Float32(0.03999999910593033))
		tmp = Float32(Float32(Float32(s * Float32(4.0)) + Float32(Float32(Float32(Float32(u * u) * Float32(Float32(Float32(Float32(Float32(8.0) / u) + Float32(21.333333333333332)) / u) + Float32(64.0))) * u) * s)) * u);
	else
		tmp = Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(u * Float32(4.0))))) * s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if ((u * single(4.0)) <= single(0.03999999910593033))
		tmp = ((s * single(4.0)) + ((((u * u) * ((((single(8.0) / u) + single(21.333333333333332)) / u) + single(64.0))) * u) * s)) * u;
	else
		tmp = log((single(1.0) / (single(1.0) - (u * single(4.0))))) * s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 4 binary32) u) < 0.0399999991

   1. Initial program 55.6%

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

   3. Taylor expanded in u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in u around -inf

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

   10. Applied rewrites 99.3%

       \[\leadsto \left(\left(\left(-\left(-64 - \frac{\frac{8}{u} + 21.333333333333332}{u}\right)\right) \cdot {u}^{3}\right) \cdot s + 4 \cdot s\right) \cdot u \]
   11. Step-by-step derivation

   12. Applied rewrites 99.3%

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

   if 0.0399999991 < (*.f32 #s(literal 4 binary32) u)

   1. Initial program 95.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.03999999910593033:\\ \;\;\;\;\left(s \cdot 4 + \left(\left(\left(u \cdot u\right) \cdot \left(\frac{\frac{8}{u} + 21.333333333333332}{u} + 64\right)\right) \cdot u\right) \cdot s\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((s * 4.0e0) + ((((u * u) * ((((8.0e0 / u) + 21.333333333333332e0) / u) + 64.0e0)) * u) * s)) * u
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = ((s * single(4.0)) + ((((u * u) * ((((single(8.0) / u) + single(21.333333333333332)) / u) + single(64.0))) * u) * s)) * u;
end

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 70.4%

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

7. Applied rewrites 84.8%

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

8. Taylor expanded in u around -inf

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

10. Applied rewrites 92.9%

    \[\leadsto \left(\left(\left(-\left(-64 - \frac{\frac{8}{u} + 21.333333333333332}{u}\right)\right) \cdot {u}^{3}\right) \cdot s + 4 \cdot s\right) \cdot u \]
11. Step-by-step derivation

12. Applied rewrites 92.9%

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

13. Final simplification 92.9%

    \[\leadsto \left(s \cdot 4 + \left(\left(\left(u \cdot u\right) \cdot \left(\frac{\frac{8}{u} + 21.333333333333332}{u} + 64\right)\right) \cdot u\right) \cdot s\right) \cdot u \]
14. Add Preprocessing

Alternative 3: 93.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((((((21.333333333333332e0 / u) + ((8.0e0 / (u * u)) + 64.0e0)) * (u * u)) * u) + 4.0e0) * s) * u
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = ((((((single(21.333333333333332) / u) + ((single(8.0) / (u * u)) + single(64.0))) * (u * u)) * u) + single(4.0)) * s) * u;
end

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 70.4%

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

7. Applied rewrites 84.7%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 92.6%

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

11. Final simplification 92.6%

    \[\leadsto \left(\left(\left(\left(\frac{21.333333333333332}{u} + \left(\frac{8}{u \cdot u} + 64\right)\right) \cdot \left(u \cdot u\right)\right) \cdot u + 4\right) \cdot s\right) \cdot u \]
12. Add Preprocessing

Alternative 4: 87.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 70.4%

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

7. Applied rewrites 84.8%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 84.8%

    \[\leadsto \left(\left(8 \cdot u\right) \cdot s + 4 \cdot s\right) \cdot u \]

11. Final simplification 84.8%

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

Alternative 5: 87.0% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 70.4%

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

7. Applied rewrites 84.7%

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

8. Taylor expanded in u around 0

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

10. Applied rewrites 84.7%

    \[\leadsto \left(s \cdot \left(8 \cdot u + 4\right)\right) \cdot u \]

11. Final simplification 84.7%

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

Alternative 6: 74.1% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.3%

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

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

   2. lower-*.f32 70.4

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

5. Applied rewrites 70.4%

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

6. Final simplification 70.4%

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

Alternative 7: 73.8% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 70.4%

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

7. Applied rewrites 84.8%

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

8. Taylor expanded in u around -inf

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

10. Applied rewrites 92.9%

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

11. Taylor expanded in u around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 70.3

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

13. Applied rewrites 70.3%

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

14. Final simplification 70.3%

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

Reproduce

herbie shell --seed 2024242 
(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))))))