Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.7% → 99.4%

Time: 22.9s

Alternatives: 10

Speedup: 21.8×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log1p (* u -4.0)) (- s)))

C

float code(float s, float u) {
	return log1pf((u * -4.0f)) * -s;
}

Julia

function code(s, u)
	return Float32(log1p(Float32(u * Float32(-4.0))) * Float32(-s))
end

Derivation

1. Initial program 59.7%

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

   1. *-commutative 59.7%

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

   2. log-rec 62.4%

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

   3. distribute-lft-neg-out 62.4%

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

   4. distribute-rgt-neg-in 62.4%

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

   5. sub-neg 62.4%

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

   6. log1p-def 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 91.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 90.1%

   \[\leadsto \color{blue}{21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + \left(4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 90.1%

      \[\leadsto \color{blue}{\left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 4 \cdot \left(s \cdot u\right)\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]

   2. +-commutative 90.1%

      \[\leadsto \color{blue}{\left(4 \cdot \left(s \cdot u\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right)} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   3. associate-+l+ 90.2%

      \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + \left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right)} \]

   4. *-commutative 90.2%

      \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} + \left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   5. associate-*l* 90.3%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4\right)} + \left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   6. *-commutative 90.3%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\color{blue}{\left(s \cdot {u}^{3}\right) \cdot 21.333333333333332} + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   7. associate-*l* 90.3%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\color{blue}{s \cdot \left({u}^{3} \cdot 21.333333333333332\right)} + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   8. *-commutative 90.3%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(s \cdot \left({u}^{3} \cdot 21.333333333333332\right) + \color{blue}{\left(s \cdot {u}^{2}\right) \cdot 8}\right) \]

   9. associate-*l* 90.4%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(s \cdot \left({u}^{3} \cdot 21.333333333333332\right) + \color{blue}{s \cdot \left({u}^{2} \cdot 8\right)}\right) \]

   10. distribute-lft-out 90.4%

       \[\leadsto s \cdot \left(u \cdot 4\right) + \color{blue}{s \cdot \left({u}^{3} \cdot 21.333333333333332 + {u}^{2} \cdot 8\right)} \]

   11. distribute-lft-out 90.5%

       \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left({u}^{3} \cdot 21.333333333333332 + {u}^{2} \cdot 8\right)\right)} \]

   12. *-commutative 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{21.333333333333332 \cdot {u}^{3}} + {u}^{2} \cdot 8\right)\right) \]

   13. cube-mult 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(21.333333333333332 \cdot \color{blue}{\left(u \cdot \left(u \cdot u\right)\right)} + {u}^{2} \cdot 8\right)\right) \]

   14. unpow2 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(21.333333333333332 \cdot \left(u \cdot \color{blue}{{u}^{2}}\right) + {u}^{2} \cdot 8\right)\right) \]

   15. associate-*r* 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{\left(21.333333333333332 \cdot u\right) \cdot {u}^{2}} + {u}^{2} \cdot 8\right)\right) \]

   16. *-commutative 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(21.333333333333332 \cdot u\right) \cdot {u}^{2} + \color{blue}{8 \cdot {u}^{2}}\right)\right) \]

   17. distribute-rgt-out 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{{u}^{2} \cdot \left(21.333333333333332 \cdot u + 8\right)}\right) \]

   18. unpow2 90.5%

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

6. Simplified 90.5%

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

7. Final simplification 90.5%

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

Alternative 3: 90.9% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 90.5%

   \[\leadsto s \cdot \left(-\color{blue}{\left(-4 \cdot u + \left(-8 \cdot {u}^{2} + -21.333333333333332 \cdot {u}^{3}\right)\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 90.5%

      \[\leadsto s \cdot \left(-\color{blue}{\left(\left(-8 \cdot {u}^{2} + -21.333333333333332 \cdot {u}^{3}\right) + -4 \cdot u\right)}\right) \]

   2. +-commutative 90.5%

      \[\leadsto s \cdot \left(-\left(\color{blue}{\left(-21.333333333333332 \cdot {u}^{3} + -8 \cdot {u}^{2}\right)} + -4 \cdot u\right)\right) \]

   3. *-commutative 90.5%

      \[\leadsto s \cdot \left(-\left(\left(\color{blue}{{u}^{3} \cdot -21.333333333333332} + -8 \cdot {u}^{2}\right) + -4 \cdot u\right)\right) \]

   4. cube-mult 90.5%

      \[\leadsto s \cdot \left(-\left(\left(\color{blue}{\left(u \cdot \left(u \cdot u\right)\right)} \cdot -21.333333333333332 + -8 \cdot {u}^{2}\right) + -4 \cdot u\right)\right) \]

   5. unpow2 90.5%

      \[\leadsto s \cdot \left(-\left(\left(\left(u \cdot \color{blue}{{u}^{2}}\right) \cdot -21.333333333333332 + -8 \cdot {u}^{2}\right) + -4 \cdot u\right)\right) \]

   6. associate-*l* 90.5%

      \[\leadsto s \cdot \left(-\left(\left(\color{blue}{u \cdot \left({u}^{2} \cdot -21.333333333333332\right)} + -8 \cdot {u}^{2}\right) + -4 \cdot u\right)\right) \]

   7. *-commutative 90.5%

      \[\leadsto s \cdot \left(-\left(\left(u \cdot \left({u}^{2} \cdot -21.333333333333332\right) + \color{blue}{{u}^{2} \cdot -8}\right) + -4 \cdot u\right)\right) \]

   8. unpow2 90.5%

      \[\leadsto s \cdot \left(-\left(\left(u \cdot \left({u}^{2} \cdot -21.333333333333332\right) + \color{blue}{\left(u \cdot u\right)} \cdot -8\right) + -4 \cdot u\right)\right) \]

   9. associate-*l* 90.5%

      \[\leadsto s \cdot \left(-\left(\left(u \cdot \left({u}^{2} \cdot -21.333333333333332\right) + \color{blue}{u \cdot \left(u \cdot -8\right)}\right) + -4 \cdot u\right)\right) \]

   10. distribute-lft-out 90.5%

       \[\leadsto s \cdot \left(-\left(\color{blue}{u \cdot \left({u}^{2} \cdot -21.333333333333332 + u \cdot -8\right)} + -4 \cdot u\right)\right) \]

   11. *-commutative 90.5%

       \[\leadsto s \cdot \left(-\left(u \cdot \left({u}^{2} \cdot -21.333333333333332 + u \cdot -8\right) + \color{blue}{u \cdot -4}\right)\right) \]

   12. distribute-lft-out 90.1%

       \[\leadsto s \cdot \left(-\color{blue}{u \cdot \left(\left({u}^{2} \cdot -21.333333333333332 + u \cdot -8\right) + -4\right)}\right) \]

   13. unpow2 90.1%

       \[\leadsto s \cdot \left(-u \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot -21.333333333333332 + u \cdot -8\right) + -4\right)\right) \]

   14. associate-*l* 90.1%

       \[\leadsto s \cdot \left(-u \cdot \left(\left(\color{blue}{u \cdot \left(u \cdot -21.333333333333332\right)} + u \cdot -8\right) + -4\right)\right) \]

   15. distribute-lft-out 90.1%

       \[\leadsto s \cdot \left(-u \cdot \left(\color{blue}{u \cdot \left(u \cdot -21.333333333333332 + -8\right)} + -4\right)\right) \]

6. Simplified 90.1%

   \[\leadsto s \cdot \left(-\color{blue}{u \cdot \left(u \cdot \left(u \cdot -21.333333333333332 + -8\right) + -4\right)}\right) \]

7. Final simplification 90.1%

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

Alternative 4: 86.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 90.1%

   \[\leadsto \color{blue}{21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + \left(4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 90.1%

      \[\leadsto \color{blue}{\left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 4 \cdot \left(s \cdot u\right)\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]

   2. +-commutative 90.1%

      \[\leadsto \color{blue}{\left(4 \cdot \left(s \cdot u\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right)} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   3. associate-+l+ 90.2%

      \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + \left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right)} \]

   4. *-commutative 90.2%

      \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} + \left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   5. associate-*l* 90.3%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4\right)} + \left(21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   6. *-commutative 90.3%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\color{blue}{\left(s \cdot {u}^{3}\right) \cdot 21.333333333333332} + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   7. associate-*l* 90.3%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\color{blue}{s \cdot \left({u}^{3} \cdot 21.333333333333332\right)} + 8 \cdot \left(s \cdot {u}^{2}\right)\right) \]

   8. *-commutative 90.3%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(s \cdot \left({u}^{3} \cdot 21.333333333333332\right) + \color{blue}{\left(s \cdot {u}^{2}\right) \cdot 8}\right) \]

   9. associate-*l* 90.4%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(s \cdot \left({u}^{3} \cdot 21.333333333333332\right) + \color{blue}{s \cdot \left({u}^{2} \cdot 8\right)}\right) \]

   10. distribute-lft-out 90.4%

       \[\leadsto s \cdot \left(u \cdot 4\right) + \color{blue}{s \cdot \left({u}^{3} \cdot 21.333333333333332 + {u}^{2} \cdot 8\right)} \]

   11. distribute-lft-out 90.5%

       \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left({u}^{3} \cdot 21.333333333333332 + {u}^{2} \cdot 8\right)\right)} \]

   12. *-commutative 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{21.333333333333332 \cdot {u}^{3}} + {u}^{2} \cdot 8\right)\right) \]

   13. cube-mult 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(21.333333333333332 \cdot \color{blue}{\left(u \cdot \left(u \cdot u\right)\right)} + {u}^{2} \cdot 8\right)\right) \]

   14. unpow2 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(21.333333333333332 \cdot \left(u \cdot \color{blue}{{u}^{2}}\right) + {u}^{2} \cdot 8\right)\right) \]

   15. associate-*r* 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{\left(21.333333333333332 \cdot u\right) \cdot {u}^{2}} + {u}^{2} \cdot 8\right)\right) \]

   16. *-commutative 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(21.333333333333332 \cdot u\right) \cdot {u}^{2} + \color{blue}{8 \cdot {u}^{2}}\right)\right) \]

   17. distribute-rgt-out 90.5%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{{u}^{2} \cdot \left(21.333333333333332 \cdot u + 8\right)}\right) \]

   18. unpow2 90.5%

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

6. Simplified 90.5%

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

7. Taylor expanded in u around 0 85.7%

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

8. Final simplification 85.7%

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

Alternative 5: 86.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 85.4%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
5. Step-by-step derivation

   1. associate-*r* 85.6%

      \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   2. *-commutative 85.6%

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s\right)} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   3. *-commutative 85.6%

      \[\leadsto u \cdot \left(4 \cdot s\right) + \color{blue}{\left(s \cdot {u}^{2}\right) \cdot 8} \]

   4. unpow2 85.6%

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

   5. associate-*r* 85.6%

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

   6. *-commutative 85.6%

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

   7. associate-*l* 85.7%

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

   8. distribute-lft-out 85.7%

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

   9. *-commutative 85.7%

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

   10. associate-*r* 85.7%

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

6. Simplified 85.7%

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

7. Final simplification 85.7%

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

Alternative 6: 86.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 85.4%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
5. Step-by-step derivation

   1. associate-*r* 85.6%

      \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   2. *-commutative 85.6%

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s\right)} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   3. *-commutative 85.6%

      \[\leadsto u \cdot \left(4 \cdot s\right) + \color{blue}{\left(s \cdot {u}^{2}\right) \cdot 8} \]

   4. unpow2 85.6%

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

   5. associate-*r* 85.6%

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

   6. *-commutative 85.6%

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

   7. associate-*l* 85.7%

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

   8. distribute-lft-out 85.7%

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

   9. *-commutative 85.7%

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

   10. associate-*r* 85.7%

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

6. Simplified 85.7%

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

7. Taylor expanded in s around 0 85.5%

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

8. Final simplification 85.5%

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

Alternative 7: 86.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 85.4%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
5. Step-by-step derivation

   1. associate-*r* 85.6%

      \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   2. *-commutative 85.6%

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s\right)} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   3. *-commutative 85.6%

      \[\leadsto u \cdot \left(4 \cdot s\right) + \color{blue}{\left(s \cdot {u}^{2}\right) \cdot 8} \]

   4. unpow2 85.6%

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

   5. associate-*r* 85.6%

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

   6. *-commutative 85.6%

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

   7. associate-*l* 85.7%

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

   8. distribute-lft-out 85.7%

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

   9. *-commutative 85.7%

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

   10. associate-*r* 85.7%

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

6. Simplified 85.7%

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

7. Taylor expanded in s around 0 85.5%

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

   1. *-commutative 85.5%

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

9. Simplified 85.5%

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

10. Final simplification 85.5%

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

Alternative 8: 73.5% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 73.0%

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

   1. *-commutative 73.0%

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

6. Simplified 73.0%

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

7. Final simplification 73.0%

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

Alternative 9: 73.8% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

4. Taylor expanded in u around 0 73.0%

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

   1. associate-*r* 73.2%

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

   2. *-commutative 73.2%

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

6. Simplified 73.2%

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

7. Final simplification 73.2%

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

Alternative 10: 16.7% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s 0.0))

C

float code(float s, float u) {
	return s * 0.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * 0.0e0
end function

Julia

function code(s, u)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(s, u)
	tmp = s * single(0.0);
end

Derivation

1. Initial program 59.7%

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

   1. log-rec 62.4%

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

   2. cancel-sign-sub-inv 62.4%

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

   3. metadata-eval 62.4%

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

3. Simplified 62.4%

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

5. Applied egg-rr 15.3%

   \[\leadsto s \cdot \left(-\color{blue}{0}\right) \]

6. Final simplification 15.3%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023252 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))