Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 99.4%

Time: 11.0s

Alternatives: 14

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 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 93.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)\\ \left(s \cdot u\right) \cdot \frac{16 - \mathsf{fma}\left(u, 21.333333333333332, 8\right) \cdot \left(u \cdot t\_0\right)}{4 - t\_0} \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (* u (fma u (fma u 64.0 21.333333333333332) 8.0))))
   (*
    (* s u)
    (/ (- 16.0 (* (fma u 21.333333333333332 8.0) (* u t_0))) (- 4.0 t_0)))))

C

float code(float s, float u) {
	float t_0 = u * fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f);
	return (s * u) * ((16.0f - (fmaf(u, 21.333333333333332f, 8.0f) * (u * t_0))) / (4.0f - t_0));
}

Julia

function code(s, u)
	t_0 = Float32(u * fma(u, fma(u, Float32(64.0), Float32(21.333333333333332)), Float32(8.0)))
	return Float32(Float32(s * u) * Float32(Float32(Float32(16.0) - Float32(fma(u, Float32(21.333333333333332), Float32(8.0)) * Float32(u * t_0))) / Float32(Float32(4.0) - t_0)))
end

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 93.0%

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

7. Applied rewrites 92.8%

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

9. Applied rewrites 92.8%

   \[\leadsto \left(u \cdot s\right) \cdot \frac{16 - \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right) \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)\right)\right)}{\color{blue}{4 - u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)}} \]

10. Taylor expanded in u around 0

    \[\leadsto \left(u \cdot s\right) \cdot \frac{16 - \mathsf{fma}\left(u, \frac{64}{3}, 8\right) \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right)\right)\right)}{4 - u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right)} \]
11. Step-by-step derivation

12. Applied rewrites 93.7%

    \[\leadsto \left(u \cdot s\right) \cdot \frac{16 - \mathsf{fma}\left(u, 21.333333333333332, 8\right) \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)\right)\right)}{4 - u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)} \]

13. Final simplification 93.7%

    \[\leadsto \left(s \cdot u\right) \cdot \frac{16 - \mathsf{fma}\left(u, 21.333333333333332, 8\right) \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)\right)\right)}{4 - u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)} \]
14. Add Preprocessing

Alternative 3: 93.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 93.0%

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

7. Applied rewrites 93.3%

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

9. Applied rewrites 93.3%

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

11. Applied rewrites 93.3%

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

12. Final simplification 93.3%

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

Alternative 4: 93.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 93.0%

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

7. Applied rewrites 93.3%

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

9. Applied rewrites 93.3%

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

10. Final simplification 93.3%

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

Alternative 5: 93.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 93.0%

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

7. Applied rewrites 93.3%

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

Alternative 6: 93.1% 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 60.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 93.0%

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

7. Applied rewrites 93.3%

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

8. Final simplification 93.3%

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

Alternative 7: 91.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 93.0%

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

7. Applied rewrites 93.3%

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

9. Applied rewrites 93.3%

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

10. Taylor expanded in u around 0

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

    1. lower-*.f32 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. distribute-rgt-out N/A

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

    7. lower-*.f32 N/A

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

    8. +-commutative N/A

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

    9. *-commutative N/A

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

    10. lower-fma.f32 N/A

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

    11. lower-*.f32 91.8

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

12. Applied rewrites 91.8%

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

13. Final simplification 91.8%

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

Alternative 8: 91.1% 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 60.7%

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

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. associate-+r+ N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. distribute-rgt-in N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. distribute-lft-out N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. distribute-rgt-in N/A

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

5. Applied rewrites 91.3%

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

7. Applied rewrites 91.6%

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

8. Final simplification 91.6%

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

Alternative 9: 87.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 87.4

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

5. Applied rewrites 87.4%

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

7. Applied rewrites 87.7%

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

9. Applied rewrites 87.8%

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

10. Final simplification 87.8%

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

Alternative 10: 87.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 87.4

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

5. Applied rewrites 87.4%

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

7. Applied rewrites 87.7%

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

9. Applied rewrites 87.8%

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

10. Final simplification 87.8%

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

Alternative 11: 86.8% 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 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 87.7

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

5. Applied rewrites 87.7%

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

Alternative 12: 86.8% 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 60.7%

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

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-lft-out N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 87.4

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

5. Applied rewrites 87.4%

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

7. Applied rewrites 87.7%

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

8. Final simplification 87.7%

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

Alternative 13: 73.9% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 74.9

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

5. Applied rewrites 74.9%

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

Alternative 14: 73.7% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 74.7

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

5. Applied rewrites 74.7%

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

6. Final simplification 74.7%

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

Reproduce

herbie shell --seed 2024221 
(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))))))