Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 9.7s

Alternatives: 5

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: 61.5% 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

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

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(Float32(-s) * log1p(Float32(u * Float32(-4.0))))
end

Derivation

1. Initial program 57.0%

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

   1. log-rec 59.7%

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

   2. distribute-rgt-neg-out 59.7%

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

   3. distribute-lft-neg-in 59.7%

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

   4. cancel-sign-sub-inv 59.7%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 86.3% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in u around 0 88.7%

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

   1. fma-def 88.7%

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

   2. *-commutative 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-*r* 88.8%

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

   5. *-commutative 88.8%

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

   6. unpow2 88.8%

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

   7. associate-*l* 88.8%

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

4. Simplified 88.8%

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

   1. fma-udef 88.8%

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

   2. associate-*l* 88.8%

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

6. Applied egg-rr 88.8%

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

7. Final simplification 88.8%

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

Alternative 3: 73.4% 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 57.0%

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

2. Taylor expanded in u around 0 75.9%

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

   1. *-commutative 75.9%

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

4. Simplified 75.9%

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

5. Final simplification 75.9%

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

Alternative 4: 73.6% accurate, 21.8× 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 57.0%

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

2. Taylor expanded in u around 0 76.1%

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

3. Final simplification 76.1%

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

Alternative 5: 16.3% 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 57.0%

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

3. Applied egg-rr 15.2%

   \[\leadsto s \cdot \color{blue}{\left(0.5 \cdot \mathsf{log1p}\left(4 \cdot u\right) - 0.5 \cdot \mathsf{log1p}\left(4 \cdot u\right)\right)} \]
4. Step-by-step derivation

   1. +-inverses 15.2%

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

5. Simplified 15.2%

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

6. Final simplification 15.2%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023297 
(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))))))