Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 95.9%

Time: 10.3s

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

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

FPCore

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

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f + ((0.25f - u) / 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 + ((0.25e0 - u) / 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(Float32(0.25) - u) / Float32(0.75))))))
end

MATLAB

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

Derivation

1. Initial program 95.5%

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

3. Final simplification 95.5%

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

Alternative 2: 95.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 N/A

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

   6. div-inv N/A

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

   7. lower-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval 94.8

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(1 - u \cdot 1.3333333333333333\right) - \color{blue}{-0.3333333333333333}}\right) \]

4. Applied rewrites 94.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(1 - u \cdot 1.3333333333333333\right) - -0.3333333333333333}}\right) \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. associate-+r+ N/A

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

   11. lift-fma.f32 N/A

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

   12. +-commutative N/A

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

   13. lift-fma.f32 N/A

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

   14. associate-+l+ N/A

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

   15. metadata-eval N/A

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

   16. lower-+.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 95.1

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

6. Applied rewrites 95.1%

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

Alternative 3: 32.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(u + \left(\log 0.75 + u \cdot \left(u \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (+ u (+ (log 0.75) (* u (* u 0.5))))))

C

float code(float s, float u) {
	return (3.0f * s) * (u + (logf(0.75f) + (u * (u * 0.5f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * (u + (log(0.75e0) + (u * (u * 0.5e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(u + Float32(log(Float32(0.75)) + Float32(u * Float32(u * Float32(0.5))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * (u + (log(single(0.75)) + (u * (u * single(0.5)))));
end

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(u, 1 + \frac{1}{2} \cdot u, \log \frac{3}{4}\right)} \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \color{blue}{\frac{1}{2} \cdot u + 1}, \log \frac{3}{4}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \frac{1}{2}} + 1, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{1}{2}, 1\right)}, \log \frac{3}{4}\right) \]

   6. lower-log.f32 10.9

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 0.5, 1\right), \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 0.5, 1\right), \log 0.75\right)} \]
6. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \frac{1}{2}, 1\right)} + \log \frac{3}{4}\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(u \cdot \mathsf{fma}\left(u, \frac{1}{2}, 1\right) + \color{blue}{\log \frac{3}{4}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + u \cdot \mathsf{fma}\left(u, \frac{1}{2}, 1\right)\right)} \]

   4. lift-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} + u \cdot \color{blue}{\left(u \cdot \frac{1}{2} + 1\right)}\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} + \color{blue}{\left(\left(u \cdot \frac{1}{2}\right) \cdot u + 1 \cdot u\right)}\right) \]

   6. *-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} + \left(\left(u \cdot \frac{1}{2}\right) \cdot u + \color{blue}{u}\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\left(\log \frac{3}{4} + \left(u \cdot \frac{1}{2}\right) \cdot u\right) + u\right)} \]

   8. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\left(\log \frac{3}{4} + \left(u \cdot \frac{1}{2}\right) \cdot u\right) + u\right)} \]

   9. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(\log \frac{3}{4} + \left(u \cdot \frac{1}{2}\right) \cdot u\right)} + u\right) \]

   10. *-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(\log \frac{3}{4} + \color{blue}{u \cdot \left(u \cdot \frac{1}{2}\right)}\right) + u\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(\log \frac{3}{4} + \color{blue}{u \cdot \left(u \cdot \frac{1}{2}\right)}\right) + u\right) \]

   12. lower-*.f32 32.0

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(\log 0.75 + u \cdot \color{blue}{\left(u \cdot 0.5\right)}\right) + u\right) \]

7. Applied rewrites 32.0%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\left(\log 0.75 + u \cdot \left(u \cdot 0.5\right)\right) + u\right)} \]

8. Final simplification 32.0%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(u + \left(\log 0.75 + u \cdot \left(u \cdot 0.5\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 32.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(u \cdot \left(u \cdot 0.5\right) + \left(u + \log 0.75\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (+ (* u (* u 0.5)) (+ u (log 0.75)))))

C

float code(float s, float u) {
	return (3.0f * s) * ((u * (u * 0.5f)) + (u + logf(0.75f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * ((u * (u * 0.5e0)) + (u + log(0.75e0)))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(Float32(u * Float32(u * Float32(0.5))) + Float32(u + log(Float32(0.75)))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * ((u * (u * single(0.5))) + (u + log(single(0.75))));
end

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in u around 0

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

   1. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + \frac{1}{2} \cdot u\right) + \log \frac{3}{4}\right)} \]

   2. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(u, 1 + \frac{1}{2} \cdot u, \log \frac{3}{4}\right)} \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \color{blue}{\frac{1}{2} \cdot u + 1}, \log \frac{3}{4}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \frac{1}{2}} + 1, \log \frac{3}{4}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{1}{2}, 1\right)}, \log \frac{3}{4}\right) \]

   6. lower-log.f32 10.9

      \[\leadsto \left(3 \cdot s\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 0.5, 1\right), \color{blue}{\log 0.75}\right) \]

5. Applied rewrites 10.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 0.5, 1\right), \log 0.75\right)} \]
6. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \frac{1}{2}, 1\right)} + \log \frac{3}{4}\right) \]

   2. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(u \cdot \mathsf{fma}\left(u, \frac{1}{2}, 1\right) + \color{blue}{\log \frac{3}{4}}\right) \]

   3. lift-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(u \cdot \color{blue}{\left(u \cdot \frac{1}{2} + 1\right)} + \log \frac{3}{4}\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\left(\left(u \cdot \frac{1}{2}\right) \cdot u + 1 \cdot u\right)} + \log \frac{3}{4}\right) \]

   5. *-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\left(\left(u \cdot \frac{1}{2}\right) \cdot u + \color{blue}{u}\right) + \log \frac{3}{4}\right) \]

   6. associate-+l+ N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\left(u \cdot \frac{1}{2}\right) \cdot u + \left(u + \log \frac{3}{4}\right)\right)} \]

   7. lower-+.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\left(u \cdot \frac{1}{2}\right) \cdot u + \left(u + \log \frac{3}{4}\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{u \cdot \left(u \cdot \frac{1}{2}\right)} + \left(u + \log \frac{3}{4}\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{u \cdot \left(u \cdot \frac{1}{2}\right)} + \left(u + \log \frac{3}{4}\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(u \cdot \color{blue}{\left(u \cdot \frac{1}{2}\right)} + \left(u + \log \frac{3}{4}\right)\right) \]

   11. lower-+.f32 32.0

       \[\leadsto \left(3 \cdot s\right) \cdot \left(u \cdot \left(u \cdot 0.5\right) + \color{blue}{\left(u + \log 0.75\right)}\right) \]

7. Applied rewrites 32.0%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(u \cdot 0.5\right) + \left(u + \log 0.75\right)\right)} \]
8. Add Preprocessing

Alternative 5: 29.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot -1.3333333333333333\right) \cdot \left(s \cdot -3\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* u -1.3333333333333333) (* s -3.0)))

C

float code(float s, float u) {
	return (u * -1.3333333333333333f) * (s * -3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (u * (-1.3333333333333333e0)) * (s * (-3.0e0))
end function

Julia

function code(s, u)
	return Float32(Float32(u * Float32(-1.3333333333333333)) * Float32(s * Float32(-3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = (u * single(-1.3333333333333333)) * (s * single(-3.0));
end

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. log-rec N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. distribute-lft-neg-in N/A

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

   5. distribute-lft-neg-in N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(\frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)} \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \left(u - \frac{1}{4}\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3}} \cdot \left(u - \frac{1}{4}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right) \]

   15. distribute-lft-in N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \color{blue}{\frac{-1}{4}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \color{blue}{\frac{1}{3}}\right) \]

   18. lower-fma.f32 7.8

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)}\right) \]

5. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u + \frac{1}{3}}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot \frac{-4}{3}} + \frac{1}{3}\right) \]

   3. lower-*.f32 9.5

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333} + 0.3333333333333333\right) \]

7. Applied rewrites 9.5%

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333 + 0.3333333333333333}\right) \]

8. Taylor expanded in u around inf

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u}\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 29.7

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333}\right) \]

10. Applied rewrites 29.7%

    \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333}\right) \]

11. Taylor expanded in u around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 29.7

       \[\leadsto \left(s \cdot -3\right) \cdot \color{blue}{\left(u \cdot -1.3333333333333333\right)} \]

13. Applied rewrites 29.7%

    \[\leadsto \left(s \cdot -3\right) \cdot \color{blue}{\left(u \cdot -1.3333333333333333\right)} \]

14. Final simplification 29.7%

    \[\leadsto \left(u \cdot -1.3333333333333333\right) \cdot \left(s \cdot -3\right) \]
15. Add Preprocessing

Alternative 6: 29.6% accurate, 12.6× speedup

Math

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

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(s * Float32(u * Float32(4.0)))
end

MATLAB

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. log-rec N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. distribute-lft-neg-in N/A

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

   5. distribute-lft-neg-in N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(\frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)} \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) \cdot \left(u - \frac{1}{4}\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3}} \cdot \left(u - \frac{1}{4}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)}\right) \]

   15. distribute-lft-in N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \frac{-4}{3} \cdot \color{blue}{\frac{-1}{4}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{-4}{3} \cdot u + \color{blue}{\frac{1}{3}}\right) \]

   18. lower-fma.f32 7.8

       \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)}\right) \]

5. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u + \frac{1}{3}}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot \frac{-4}{3}} + \frac{1}{3}\right) \]

   3. lower-*.f32 9.5

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333} + 0.3333333333333333\right) \]

7. Applied rewrites 9.5%

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333 + 0.3333333333333333}\right) \]

8. Taylor expanded in u around inf

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot u}\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 29.7

      \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333}\right) \]

10. Applied rewrites 29.7%

    \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\color{blue}{u \cdot -1.3333333333333333}\right) \]

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. associate-*l* N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 29.7

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

13. Applied rewrites 29.7%

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

Reproduce

herbie shell --seed 2024216 
(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))))))