Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 95.9%

Time: 9.8s

Alternatives: 9

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 96.1%

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

3. Final simplification 96.1%

   \[\leadsto \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \cdot \left(s \cdot 3\right) \]
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 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.1

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

4. Applied rewrites 96.1%

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

5. Final simplification 96.1%

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

Alternative 3: 95.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return logf((1.0f / (1.3333333333333333f - (1.3333333333333333f * u)))) * (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 - (1.3333333333333333e0 * u)))) * (s * 3.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.1%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.1

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

4. Applied rewrites 96.1%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333 \cdot \left(u - 0.25\right) + 1}}\right) \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

   6. div-inv N/A

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

   7. clear-num N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 95.9

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{-1}{\color{blue}{\frac{0.75}{u - 0.25}}} + 1}\right) \]

6. Applied rewrites 95.9%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{-1}{\frac{0.75}{u - 0.25}}} + 1}\right) \]
7. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift--.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. clear-num N/A

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

   8. neg-mul-1 N/A

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

   9. sub-neg N/A

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

   10. div-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. associate--r+ N/A

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

   18. metadata-eval N/A

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

   19. lower--.f32 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f32 95.5

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

8. Applied rewrites 95.5%

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

9. Final simplification 95.5%

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

Alternative 4: 9.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right) \cdot \left(-s\right)\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (log1p (* -1.3333333333333333 (- u 0.25))) (- s)) 3.0))

C

float code(float s, float u) {
	return (log1pf((-1.3333333333333333f * (u - 0.25f))) * -s) * 3.0f;
}

Julia

function code(s, u)
	return Float32(Float32(log1p(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25)))) * Float32(-s)) * Float32(3.0))
end

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 9.4%

   \[\leadsto \color{blue}{\left(\left(-s\right) \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot 3} \]

5. Final simplification 9.4%

   \[\leadsto \left(\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right) \cdot \left(-s\right)\right) \cdot 3 \]
6. Add Preprocessing

Alternative 5: 9.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* -3.0 (log1p (* -1.3333333333333333 (- u 0.25)))) s))

C

float code(float s, float u) {
	return (-3.0f * log1pf((-1.3333333333333333f * (u - 0.25f)))) * s;
}

Julia

function code(s, u)
	return Float32(Float32(Float32(-3.0) * log1p(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25))))) * s)
end

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 9.3%

   \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s} \]
5. Add Preprocessing

Alternative 6: 28.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 0.6666666666666666 \cdot s\right) \cdot -3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (log 0.6666666666666666) s) -3.0))

C

float code(float s, float u) {
	return (logf(0.6666666666666666f) * s) * -3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log(0.6666666666666666e0) * s) * (-3.0e0)
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(0.6666666666666666)) * s) * Float32(-3.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (log(single(0.6666666666666666)) * s) * single(-3.0);
end

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

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

   2. +-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 7.4

       \[\leadsto \mathsf{fma}\left(s \cdot 3, \color{blue}{0}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)}\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

   16. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

4. Applied rewrites 33.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot 3, 0, \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right)} \]

6. Applied rewrites 10.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) \cdot -3, 0\right)} \]

7. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-log.f32 27.8

      \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

9. Applied rewrites 27.8%

   \[\leadsto \color{blue}{\left(\log 0.6666666666666666 \cdot s\right) \cdot -3} \]
10. Add Preprocessing

Alternative 7: 26.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

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

   2. +-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 7.7

       \[\leadsto \mathsf{fma}\left(s \cdot 3, \color{blue}{0}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)}\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

   16. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

4. Applied rewrites 33.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot 3, 0, \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right)} \]

5. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. +-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 7.1

       \[\leadsto \mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \color{blue}{\mathsf{fma}\left(1.5, u, 3\right)}\right) \cdot u\right) \]

7. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \mathsf{fma}\left(1.5, u, 3\right)\right) \cdot u\right)} \]

8. Taylor expanded in u around inf

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

10. Applied rewrites 26.1%

    \[\leadsto \left(\left(u \cdot u\right) \cdot s\right) \cdot \color{blue}{1.5} \]
11. Step-by-step derivation

12. Applied rewrites 26.1%

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

Alternative 8: 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 96.1%

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

   1. lift-*.f32 N/A

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

   2. +-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 7.4

       \[\leadsto \mathsf{fma}\left(s \cdot 3, \color{blue}{0}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)}\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

   16. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

4. Applied rewrites 33.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot 3, 0, \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right)} \]

5. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. +-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 7.1

       \[\leadsto \mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \color{blue}{\mathsf{fma}\left(1.5, u, 3\right)}\right) \cdot u\right) \]

7. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \mathsf{fma}\left(1.5, u, 3\right)\right) \cdot u\right)} \]

8. Taylor expanded in u around inf

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

10. Applied rewrites 26.1%

    \[\leadsto \left(\left(u \cdot u\right) \cdot s\right) \cdot \color{blue}{1.5} \]
11. Step-by-step derivation

12. Applied rewrites 26.1%

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

Alternative 9: 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 96.1%

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

   1. lift-*.f32 N/A

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

   2. +-lft-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(0 + \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)} \]

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval 7.4

       \[\leadsto \mathsf{fma}\left(s \cdot 3, \color{blue}{0}, \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)}\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, 3 \cdot \color{blue}{\left(\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) \cdot s\right)}\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

   16. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(s \cdot 3, 0, \color{blue}{\left(3 \cdot \log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) \cdot s}\right) \]

4. Applied rewrites 33.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot 3, 0, \left(-3 \cdot \mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right)} \]

5. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-out N/A

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

   12. +-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 7.1

       \[\leadsto \mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \color{blue}{\mathsf{fma}\left(1.5, u, 3\right)}\right) \cdot u\right) \]

7. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-3 \cdot s, \log 1.3333333333333333, \left(s \cdot \mathsf{fma}\left(1.5, u, 3\right)\right) \cdot u\right)} \]

8. Taylor expanded in u around inf

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

10. Applied rewrites 26.1%

    \[\leadsto \left(\left(u \cdot u\right) \cdot s\right) \cdot \color{blue}{1.5} \]
11. Step-by-step derivation

12. Applied rewrites 26.1%

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

13. Final simplification 26.1%

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

Reproduce

herbie shell --seed 2024249 
(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))))))