Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 99.4%

Time: 10.6s

Alternatives: 13

Speedup: 11.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 61.4% 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, 1.1× 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 59.3%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. log-rec N/A

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

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

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

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

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

   5. lower-*.f32 N/A

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

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

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

   7. metadata-eval N/A

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

   8. lower-log1p.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 99.4

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

5. Applied rewrites 99.4%

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

Alternative 2: 93.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return s * fmaf(fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f), (u * u), (u * 4.0f));
}

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), 4\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, 4\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{64}{3} + 64 \cdot u, 8\right)}, 4\right)\right) \]

   6. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{64 \cdot u + \frac{64}{3}}, 8\right), 4\right)\right) \]

   7. *-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{u \cdot 64} + \frac{64}{3}, 8\right), 4\right)\right) \]

   8. lower-fma.f32 93.6

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

5. Applied rewrites 93.6%

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

   1. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 64, \frac{64}{3}\right)} + 8\right) + 4\right)\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right)} + 4\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right)\right) \cdot u + 4 \cdot u\right)} \]

   4. *-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right) \cdot u\right)} \cdot u + 4 \cdot u\right) \]

   5. associate-*l* N/A

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right) \cdot \left(u \cdot u\right)} + 4 \cdot u\right) \]

   6. *-commutative N/A

      \[\leadsto s \cdot \left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right) \cdot \left(u \cdot u\right) + \color{blue}{u \cdot 4}\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), u \cdot u, u \cdot 4\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto s \cdot \mathsf{fma}\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), \color{blue}{u \cdot u}, u \cdot 4\right) \]

   9. lower-*.f32 93.9

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

7. Applied rewrites 93.9%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), u \cdot u, u \cdot 4\right)} \]
8. Add Preprocessing

Alternative 3: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return u * (s * fmaf(u, fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f), 4.0f));
}

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), 4\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, 4\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{64}{3} + 64 \cdot u, 8\right)}, 4\right)\right) \]

   6. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{64 \cdot u + \frac{64}{3}}, 8\right), 4\right)\right) \]

   7. *-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{u \cdot 64} + \frac{64}{3}, 8\right), 4\right)\right) \]

   8. lower-fma.f32 93.6

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

5. Applied rewrites 93.6%

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

   1. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 64, \frac{64}{3}\right)} + 8\right) + 4\right)\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right)} + 4\right)\right) \]

   3. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), 4\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), 4\right) \cdot u\right)} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), 4\right)\right) \cdot u} \]

   6. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), 4\right)\right) \cdot u} \]

   7. lower-*.f32 93.7

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

7. Applied rewrites 93.7%

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

8. Final simplification 93.7%

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

Alternative 4: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return s * (u * fmaf(u, fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f), 4.0f));
}

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\left(u \cdot \left(8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right)\right) + 4\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 8 + u \cdot \left(\frac{64}{3} + 64 \cdot u\right), 4\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \left(\frac{64}{3} + 64 \cdot u\right) + 8}, 4\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{\mathsf{fma}\left(u, \frac{64}{3} + 64 \cdot u, 8\right)}, 4\right)\right) \]

   6. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{64 \cdot u + \frac{64}{3}}, 8\right), 4\right)\right) \]

   7. *-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \color{blue}{u \cdot 64} + \frac{64}{3}, 8\right), 4\right)\right) \]

   8. lower-fma.f32 93.6

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

5. Applied rewrites 93.6%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), 4\right)\right)} \]
6. Add Preprocessing

Alternative 5: 91.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\left(u \cdot \left(8 + \frac{64}{3} \cdot u\right) + 4\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 8 + \frac{64}{3} \cdot u, 4\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{\frac{64}{3} \cdot u + 8}, 4\right)\right) \]

   5. *-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \frac{64}{3}} + 8, 4\right)\right) \]

   6. lower-fma.f32 92.0

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

5. Applied rewrites 92.0%

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

   1. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \frac{64}{3}, 8\right)} + 4\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto s \cdot \color{blue}{\left(\left(u \cdot \mathsf{fma}\left(u, \frac{64}{3}, 8\right)\right) \cdot u + 4 \cdot u\right)} \]

   3. *-commutative N/A

      \[\leadsto s \cdot \left(\left(u \cdot \mathsf{fma}\left(u, \frac{64}{3}, 8\right)\right) \cdot u + \color{blue}{u \cdot 4}\right) \]

   4. *-commutative N/A

      \[\leadsto s \cdot \left(\color{blue}{\left(\mathsf{fma}\left(u, \frac{64}{3}, 8\right) \cdot u\right)} \cdot u + u \cdot 4\right) \]

   5. associate-*l* N/A

      \[\leadsto s \cdot \left(\color{blue}{\mathsf{fma}\left(u, \frac{64}{3}, 8\right) \cdot \left(u \cdot u\right)} + u \cdot 4\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(u, \frac{64}{3}, 8\right), u \cdot u, u \cdot 4\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto s \cdot \mathsf{fma}\left(\mathsf{fma}\left(u, \frac{64}{3}, 8\right), \color{blue}{u \cdot u}, u \cdot 4\right) \]

   8. lower-*.f32 92.2

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

7. Applied rewrites 92.2%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(u, 21.333333333333332, 8\right), u \cdot u, u \cdot 4\right)} \]
8. Add Preprocessing

Alternative 6: 90.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\left(u \cdot \left(8 + \frac{64}{3} \cdot u\right) + 4\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 8 + \frac{64}{3} \cdot u, 4\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{\frac{64}{3} \cdot u + 8}, 4\right)\right) \]

   5. *-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \frac{64}{3}} + 8, 4\right)\right) \]

   6. lower-fma.f32 92.0

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

5. Applied rewrites 92.0%

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

   1. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \frac{64}{3}, 8\right)} + 4\right)\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{64}{3}, 8\right), 4\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{64}{3}, 8\right), 4\right) \cdot u\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{64}{3}, 8\right), 4\right)\right) \cdot u} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \frac{64}{3}, 8\right), 4\right)\right) \cdot u} \]

   6. lower-*.f32 92.0

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

7. Applied rewrites 92.0%

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

8. Final simplification 92.0%

   \[\leadsto u \cdot \left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 21.333333333333332, 8\right), 4\right)\right) \]
9. Add Preprocessing

Alternative 7: 90.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + \frac{64}{3} \cdot u\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\left(u \cdot \left(8 + \frac{64}{3} \cdot u\right) + 4\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 8 + \frac{64}{3} \cdot u, 4\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{\frac{64}{3} \cdot u + 8}, 4\right)\right) \]

   5. *-commutative N/A

      \[\leadsto s \cdot \left(u \cdot \mathsf{fma}\left(u, \color{blue}{u \cdot \frac{64}{3}} + 8, 4\right)\right) \]

   6. lower-fma.f32 92.0

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

5. Applied rewrites 92.0%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 21.333333333333332, 8\right), 4\right)\right)} \]
6. Add Preprocessing

Alternative 8: 86.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 88.2

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

5. Applied rewrites 88.2%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 88.3

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

7. Applied rewrites 88.3%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 88.4

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

9. Applied rewrites 88.4%

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

10. Final simplification 88.4%

    \[\leadsto u \cdot \mathsf{fma}\left(u \cdot s, 8, s \cdot 4\right) \]
11. Add Preprocessing

Alternative 9: 86.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 88.2

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

5. Applied rewrites 88.2%

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 88.4

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

7. Applied rewrites 88.4%

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

Alternative 10: 86.6% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 88.2

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

5. Applied rewrites 88.2%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 88.3

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

7. Applied rewrites 88.3%

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

8. Final simplification 88.3%

   \[\leadsto u \cdot \left(s \cdot \mathsf{fma}\left(u, 8, 4\right)\right) \]
9. Add Preprocessing

Alternative 11: 86.6% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 88.2

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

5. Applied rewrites 88.2%

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

Alternative 12: 73.8% accurate, 11.4× 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 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 75.7

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

5. Applied rewrites 75.7%

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

6. Final simplification 75.7%

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

Alternative 13: 73.6% accurate, 11.4× 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 59.3%

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

3. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 75.4

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

5. Applied rewrites 75.4%

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

Reproduce

herbie shell --seed 2024219 
(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))))))