Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.6% → 99.4%

Time: 10.8s

Alternatives: 7

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: 60.6% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.9%

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

   1. log-rec 61.0%

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

   2. sub-neg 61.0%

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

   3. distribute-rgt-neg-out 61.0%

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

   4. log1p-def 99.4%

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

   5. distribute-rgt-neg-in 99.4%

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

   6. distribute-lft-neg-in 99.4%

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

   7. *-commutative 99.4%

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

   8. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 91.2% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

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 + (u * 21.333333333333332e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

2. Taylor expanded in u around 0 92.2%

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

   1. *-commutative 92.2%

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

   2. *-commutative 92.2%

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

   3. unpow2 92.2%

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

   4. associate-*l* 92.2%

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

   5. unpow3 92.2%

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

   6. unpow2 92.2%

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

   7. associate-*r* 92.2%

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

   8. *-commutative 92.2%

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

   9. distribute-lft-out 92.2%

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

   10. distribute-lft-out 91.9%

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

   11. *-commutative 91.9%

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

   12. unpow2 91.9%

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

   13. associate-*l* 91.9%

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

   14. distribute-lft-out 91.9%

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

4. Simplified 91.9%

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

5. Final simplification 91.9%

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

Alternative 3: 87.1% 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 58.9%

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

2. Taylor expanded in u around 0 88.3%

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

   1. *-commutative 88.3%

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

   2. associate-*r* 88.4%

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

   3. *-commutative 88.4%

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

   4. associate-*r* 88.5%

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

   5. distribute-rgt-out 88.4%

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

   6. *-commutative 88.4%

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

   7. *-commutative 88.4%

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

   8. *-commutative 88.4%

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

   9. unpow2 88.4%

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

   10. associate-*l* 88.4%

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

   11. distribute-lft-out 88.3%

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

4. Simplified 88.3%

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

5. Final simplification 88.3%

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

Alternative 4: 23.1% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 4.5000001435742244 \cdot 10^{-32}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;s \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (if (<= s 4.5000001435742244e-32) (* s 0.0) (* s 0.25)))

C

float code(float s, float u) {
	float tmp;
	if (s <= 4.5000001435742244e-32f) {
		tmp = s * 0.0f;
	} else {
		tmp = s * 0.25f;
	}
	return tmp;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: tmp
    if (s <= 4.5000001435742244e-32) then
        tmp = s * 0.0e0
    else
        tmp = s * 0.25e0
    end if
    code = tmp
end function

Julia

function code(s, u)
	tmp = Float32(0.0)
	if (s <= Float32(4.5000001435742244e-32))
		tmp = Float32(s * Float32(0.0));
	else
		tmp = Float32(s * Float32(0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, u)
	tmp = single(0.0);
	if (s <= single(4.5000001435742244e-32))
		tmp = s * single(0.0);
	else
		tmp = s * single(0.25);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 4.50000014e-32

   1. Initial program 79.5%

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

   3. Applied egg-rr 47.2%

      \[\leadsto s \cdot \color{blue}{0} \]

   if 4.50000014e-32 < s

   1. Initial program 54.9%

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

   3. Applied egg-rr 17.9%

      \[\leadsto s \cdot \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.5000001435742244 \cdot 10^{-32}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;s \cdot 0.25\\ \end{array} \]

Alternative 5: 74.2% accurate, 21.8× 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 58.9%

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

2. Taylor expanded in u around 0 75.6%

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

   1. *-commutative 75.6%

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

4. Simplified 75.6%

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

5. Final simplification 75.6%

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

Alternative 6: 74.4% 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 58.9%

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

2. Taylor expanded in u around 0 75.6%

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

   1. associate-*r* 75.7%

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

   2. *-commutative 75.7%

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

4. Simplified 75.7%

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

5. Final simplification 75.7%

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

Alternative 7: 16.6% 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 58.9%

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

3. Applied egg-rr 16.3%

   \[\leadsto s \cdot \color{blue}{0} \]

4. Final simplification 16.3%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023336 
(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))))))