Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 95.9%

Time: 8.9s

Alternatives: 6

Speedup: 1.0×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Initial Program: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Alternative 1: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \cdot \left(s \cdot 3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75)))) (* s 3.0)))

C

float code(float s, float u) {
	return logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f)))) * (s * 3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0)))) * (s * 3.0e0)
end function

Julia

function code(s, u)
	return Float32(log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))) * Float32(s * Float32(3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75))))) * (s * single(3.0));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Final simplification 95.8%

   \[\leadsto \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \cdot \left(s \cdot 3\right) \]
4. Add Preprocessing

Alternative 2: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{-1.3333333333333333 \cdot \left(u - 0.25\right) + 1}\right) \cdot \left(s \cdot 3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (log (/ 1.0 (+ (* -1.3333333333333333 (- u 0.25)) 1.0))) (* s 3.0)))

C

float code(float s, float u) {
	return logf((1.0f / ((-1.3333333333333333f * (u - 0.25f)) + 1.0f))) * (s * 3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = log((1.0e0 / (((-1.3333333333333333e0) * (u - 0.25e0)) + 1.0e0))) * (s * 3.0e0)
end function

Julia

function code(s, u)
	return Float32(log(Float32(Float32(1.0) / Float32(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25))) + Float32(1.0)))) * Float32(s * Float32(3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = log((single(1.0) / ((single(-1.3333333333333333) * (u - single(0.25))) + single(1.0)))) * (s * single(3.0));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 95.8

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333} \cdot \left(u - 0.25\right) + 1}\right) \]

4. Applied rewrites 95.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333 \cdot \left(u - 0.25\right) + 1}}\right) \]

5. Final simplification 95.8%

   \[\leadsto \log \left(\frac{1}{-1.3333333333333333 \cdot \left(u - 0.25\right) + 1}\right) \cdot \left(s \cdot 3\right) \]
6. Add Preprocessing

Alternative 3: 36.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(1.5 + u\right) \cdot u\right) \cdot u + \left(\log 0.75 + u\right) \cdot 3\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (+ (* (* (+ 1.5 u) u) u) (* (+ (log 0.75) u) 3.0)) s))

C

float code(float s, float u) {
	return ((((1.5f + u) * u) * u) + ((logf(0.75f) + u) * 3.0f)) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((((1.5e0 + u) * u) * u) + ((log(0.75e0) + u) * 3.0e0)) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(Float32(1.5) + u) * u) * u) + Float32(Float32(log(Float32(0.75)) + u) * Float32(3.0))) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = ((((single(1.5) + u) * u) * u) + ((log(single(0.75)) + u) * single(3.0))) * s;
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

      \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \color{blue}{\left(\left(3 \cdot s\right) \cdot u + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \left(3 \cdot s\right) \cdot u\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   8. associate-*l* N/A

      \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   9. *-commutative N/A

      \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot u\right)} \cdot u \]

   10. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot \left(u \cdot u\right)} \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(\color{blue}{s \cdot \frac{3}{2}} + s \cdot u\right) \cdot \left(u \cdot u\right) \]

   12. distribute-lft-out N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot \left(\frac{3}{2} + u\right)\right)} \cdot \left(u \cdot u\right) \]

   13. unpow2 N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(s \cdot \left(\frac{3}{2} + u\right)\right) \cdot \color{blue}{{u}^{2}} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(\left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + \left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

5. Applied rewrites 14.5%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 36.9%

   \[\leadsto s \cdot \left(\left(\log 0.75 + u\right) \cdot 3 + \color{blue}{\left(\left(1.5 + u\right) \cdot u\right) \cdot u}\right) \]

8. Final simplification 36.9%

   \[\leadsto \left(\left(\left(1.5 + u\right) \cdot u\right) \cdot u + \left(\log 0.75 + u\right) \cdot 3\right) \cdot s \]
9. Add Preprocessing

Alternative 4: 27.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(1.5 + u\right) \cdot s\right) \cdot u\right) \cdot u \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* (+ 1.5 u) s) u) u))

C

float code(float s, float u) {
	return (((1.5f + u) * s) * u) * u;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((1.5e0 + u) * s) * u) * u
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(Float32(1.5) + u) * s) * u) * u)
end

MATLAB

function tmp = code(s, u)
	tmp = (((single(1.5) + u) * s) * u) * u;
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

      \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \color{blue}{\left(\left(3 \cdot s\right) \cdot u + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \left(3 \cdot s\right) \cdot u\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   8. associate-*l* N/A

      \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   9. *-commutative N/A

      \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot u\right)} \cdot u \]

   10. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot \left(u \cdot u\right)} \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(\color{blue}{s \cdot \frac{3}{2}} + s \cdot u\right) \cdot \left(u \cdot u\right) \]

   12. distribute-lft-out N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot \left(\frac{3}{2} + u\right)\right)} \cdot \left(u \cdot u\right) \]

   13. unpow2 N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(s \cdot \left(\frac{3}{2} + u\right)\right) \cdot \color{blue}{{u}^{2}} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(\left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + \left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

5. Applied rewrites 14.4%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]

6. Taylor expanded in u around inf

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

8. Applied rewrites 27.7%

   \[\leadsto \left(\left(\left(1.5 + u\right) \cdot s\right) \cdot u\right) \cdot \color{blue}{u} \]
9. Add Preprocessing

Alternative 5: 26.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \left(1.5 \cdot u\right) \cdot \left(u \cdot s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* 1.5 u) (* u s)))

C

float code(float s, float u) {
	return (1.5f * u) * (u * s);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.5e0 * u) * (u * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.5) * u) * Float32(u * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.5) * u) * (u * s);
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

      \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \color{blue}{\left(\left(3 \cdot s\right) \cdot u + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \left(3 \cdot s\right) \cdot u\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   8. associate-*l* N/A

      \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   9. *-commutative N/A

      \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot u\right)} \cdot u \]

   10. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot \left(u \cdot u\right)} \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(\color{blue}{s \cdot \frac{3}{2}} + s \cdot u\right) \cdot \left(u \cdot u\right) \]

   12. distribute-lft-out N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot \left(\frac{3}{2} + u\right)\right)} \cdot \left(u \cdot u\right) \]

   13. unpow2 N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(s \cdot \left(\frac{3}{2} + u\right)\right) \cdot \color{blue}{{u}^{2}} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(\left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + \left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

5. Applied rewrites 14.3%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]

6. Taylor expanded in u around inf

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

8. Applied rewrites 27.7%

   \[\leadsto \left(\left(\left(1.5 + u\right) \cdot s\right) \cdot u\right) \cdot \color{blue}{u} \]

9. Taylor expanded in u around 0

   \[\leadsto \frac{3}{2} \cdot \left(s \cdot \color{blue}{{u}^{2}}\right) \]
10. Step-by-step derivation

11. Applied rewrites 26.5%

    \[\leadsto \left(\left(u \cdot s\right) \cdot u\right) \cdot 1.5 \]
12. Step-by-step derivation

13. Applied rewrites 26.5%

    \[\leadsto \left(s \cdot u\right) \cdot \left(1.5 \cdot u\right) \]

14. Final simplification 26.5%

    \[\leadsto \left(1.5 \cdot u\right) \cdot \left(u \cdot s\right) \]
15. Add Preprocessing

Alternative 6: 26.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(1.5 \cdot u\right) \cdot s\right) \cdot u \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (* 1.5 u) s) u))

C

float code(float s, float u) {
	return ((1.5f * u) * s) * u;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((1.5e0 * u) * s) * u
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(Float32(1.5) * u) * s) * u)
end

MATLAB

function tmp = code(s, u)
	tmp = ((single(1.5) * u) * s) * u;
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Add Preprocessing

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

      \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \color{blue}{\left(\left(3 \cdot s\right) \cdot u + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(3 \cdot \left(s \cdot \log \frac{3}{4}\right) + \left(3 \cdot s\right) \cdot u\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(3 \cdot s\right) \cdot u + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{3 \cdot \left(s \cdot u\right)} + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right) + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot u + s \cdot \log \frac{3}{4}\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(s \cdot u + s \cdot \log \frac{3}{4}\right) \cdot 3} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   7. distribute-lft-out N/A

      \[\leadsto \color{blue}{\left(s \cdot \left(u + \log \frac{3}{4}\right)\right)} \cdot 3 + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   8. associate-*l* N/A

      \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right)} + \left(u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \cdot u \]

   9. *-commutative N/A

      \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot u\right)} \cdot u \]

   10. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(\frac{3}{2} \cdot s + s \cdot u\right) \cdot \left(u \cdot u\right)} \]

   11. *-commutative N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(\color{blue}{s \cdot \frac{3}{2}} + s \cdot u\right) \cdot \left(u \cdot u\right) \]

   12. distribute-lft-out N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{\left(s \cdot \left(\frac{3}{2} + u\right)\right)} \cdot \left(u \cdot u\right) \]

   13. unpow2 N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \left(s \cdot \left(\frac{3}{2} + u\right)\right) \cdot \color{blue}{{u}^{2}} \]

   14. associate-*l* N/A

       \[\leadsto s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3\right) + \color{blue}{s \cdot \left(\left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

   15. distribute-lft-out N/A

       \[\leadsto \color{blue}{s \cdot \left(\left(u + \log \frac{3}{4}\right) \cdot 3 + \left(\frac{3}{2} + u\right) \cdot {u}^{2}\right)} \]

5. Applied rewrites 14.5%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]

6. Taylor expanded in u around inf

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

8. Applied rewrites 27.7%

   \[\leadsto \left(\left(\left(1.5 + u\right) \cdot s\right) \cdot u\right) \cdot \color{blue}{u} \]

9. Taylor expanded in u around 0

   \[\leadsto \frac{3}{2} \cdot \left(s \cdot \color{blue}{{u}^{2}}\right) \]
10. Step-by-step derivation

11. Applied rewrites 26.5%

    \[\leadsto \left(\left(u \cdot s\right) \cdot u\right) \cdot 1.5 \]
12. Step-by-step derivation

13. Applied rewrites 26.5%

    \[\leadsto u \cdot \left(s \cdot \left(1.5 \cdot \color{blue}{u}\right)\right) \]

14. Final simplification 26.5%

    \[\leadsto \left(\left(1.5 \cdot u\right) \cdot s\right) \cdot u \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))