Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.6% → 98.7%

Time: 8.2s

Alternatives: 8

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: 61.6% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.03999999910593033:\\ \;\;\;\;\left(s \cdot 4 + \left({u}^{3} \cdot \left(e^{-\log \left(\frac{u}{\frac{8}{u} + 21.333333333333332}\right)} - -64\right)\right) \cdot s\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.03999999910593033)
   (*
    (+
     (* s 4.0)
     (*
      (*
       (pow u 3.0)
       (- (exp (- (log (/ u (+ (/ 8.0 u) 21.333333333333332))))) -64.0))
      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) + ((powf(u, 3.0f) * (expf(-logf((u / ((8.0f / u) + 21.333333333333332f)))) - -64.0f)) * 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 ** 3.0e0) * (exp(-log((u / ((8.0e0 / u) + 21.333333333333332e0)))) - (-64.0e0))) * 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((u ^ Float32(3.0)) * Float32(exp(Float32(-log(Float32(u / Float32(Float32(Float32(8.0) / u) + Float32(21.333333333333332)))))) - Float32(-64.0))) * 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 ^ single(3.0)) * (exp(-log((u / ((single(8.0) / u) + single(21.333333333333332))))) - single(-64.0))) * 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 54.3%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 80.2%

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

   7. Applied rewrites 94.7%

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

   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 99.4%

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

   12. Applied rewrites 99.4%

       \[\leadsto \left(\left(\left(e^{\log \left(\frac{u}{21.333333333333332 + \frac{8}{u}}\right) \cdot -1} - -64\right) \cdot {u}^{3}\right) \cdot s + 4 \cdot s\right) \cdot u \]

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

   1. Initial program 95.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u \cdot 4 \leq 0.03999999910593033:\\ \;\;\;\;\left(s \cdot 4 + \left({u}^{3} \cdot \left(e^{-\log \left(\frac{u}{\frac{8}{u} + 21.333333333333332}\right)} - -64\right)\right) \cdot s\right) \cdot u\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1}{1 - u \cdot 4}\right) \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= (* u 4.0) 0.03999999910593033)
   (*
    (+ (* (* (+ (* (+ (* 64.0 u) 21.333333333333332) u) 8.0) u) s) (* s 4.0))
    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 = (((((((64.0f * u) + 21.333333333333332f) * u) + 8.0f) * u) * s) + (s * 4.0f)) * 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 = (((((((64.0e0 * u) + 21.333333333333332e0) * u) + 8.0e0) * u) * s) + (s * 4.0e0)) * 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(Float32(Float32(Float32(Float32(Float32(Float32(64.0) * u) + Float32(21.333333333333332)) * u) + Float32(8.0)) * u) * s) + Float32(s * Float32(4.0))) * 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 = (((((((single(64.0) * u) + single(21.333333333333332)) * u) + single(8.0)) * u) * s) + (s * single(4.0))) * 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 54.3%

      \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   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 80.2%

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

   7. Applied rewrites 94.7%

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

   9. Applied rewrites 98.4%

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

   11. Applied rewrites 99.4%

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

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

   1. Initial program 95.5%

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

4. Final simplification 98.8%

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

Alternative 3: 93.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return ((((((21.333333333333332f * u) + ((64.0f * u) * u)) + 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 = ((((((21.333333333333332e0 * u) + ((64.0e0 * u) * u)) + 8.0e0) * u) * s) + (s * 4.0e0)) * u
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.9%

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

7. Applied rewrites 86.4%

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

9. Applied rewrites 90.6%

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

11. Applied rewrites 92.6%

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

12. Final simplification 92.6%

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

Alternative 4: 93.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (((((((64.0f * u) + 21.333333333333332f) * u) + 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 = (((((((64.0e0 * u) + 21.333333333333332e0) * u) + 8.0e0) * u) * s) + (s * 4.0e0)) * u
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.9%

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

7. Applied rewrites 86.4%

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

9. Applied rewrites 90.6%

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

11. Applied rewrites 92.6%

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

12. Final simplification 92.6%

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

Alternative 5: 90.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (((((21.333333333333332f * u) + 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 = (((((21.333333333333332e0 * u) + 8.0e0) * u) * s) + (s * 4.0e0)) * u
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.9%

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

7. Applied rewrites 86.4%

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

9. Applied rewrites 90.6%

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

10. Taylor expanded in u around 0

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

12. Applied rewrites 90.6%

    \[\leadsto \left(\left(\left(21.333333333333332 \cdot u + 8\right) \cdot u\right) \cdot s + 4 \cdot s\right) \cdot u \]

13. Final simplification 90.6%

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

Alternative 6: 86.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. log-div N/A

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

   6. flip-- N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{\log 1 \cdot \log 1 - \log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)}{\color{blue}{0} + \log \left(1 - 4 \cdot u\right)} \cdot s \]

   8. +-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 42.5%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 19.5

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

7. Applied rewrites 19.5%

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

9. Applied rewrites 86.4%

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

Alternative 7: 86.5% 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 60.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. log-div N/A

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

   6. flip-- N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{\log 1 \cdot \log 1 - \log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)}{\color{blue}{0} + \log \left(1 - 4 \cdot u\right)} \cdot s \]

   8. +-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 42.5%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 19.5

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

7. Applied rewrites 19.5%

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

9. Applied rewrites 27.5%

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

11. Applied rewrites 86.1%

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

12. Final simplification 86.1%

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

Alternative 8: 73.7% 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 60.9%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 72.9

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

5. Applied rewrites 72.9%

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

6. Final simplification 72.9%

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

Reproduce

herbie shell --seed 2024241 
(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))))))