Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 99.3%

Time: 8.4s

Alternatives: 4

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: 61.2% 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: 99.3% 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 59.0%

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

   1. log-rec 61.4%

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

   2. cancel-sign-sub-inv 61.4%

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

   3. log1p-def 99.4%

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

   4. *-commutative 99.4%

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

   5. 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: 86.8% accurate, 10.9× speedup

Math

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

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(-Float32(-4.0)) - Float32(u * Float32(-8.0)))))
end

MATLAB

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

Derivation

1. Initial program 59.0%

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

   1. log-rec 61.4%

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

   2. cancel-sign-sub-inv 61.4%

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

   3. log1p-def 99.4%

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

   4. *-commutative 99.4%

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

   5. 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. Taylor expanded in u around 0 87.8%

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

   1. unpow2 87.8%

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

   2. associate-*r* 87.8%

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

   3. distribute-rgt-out 87.5%

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

   4. *-commutative 87.5%

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

6. Simplified 87.5%

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

7. Final simplification 87.5%

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

Alternative 3: 73.7% 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 59.0%

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

   1. log-rec 61.4%

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

   2. cancel-sign-sub-inv 61.4%

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

   3. log1p-def 99.4%

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

   4. *-commutative 99.4%

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

   5. 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. Taylor expanded in u around 0 75.5%

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

5. Final simplification 75.5%

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

Alternative 4: 73.9% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.0%

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

   1. log-rec 61.4%

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

   2. cancel-sign-sub-inv 61.4%

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

   3. log1p-def 99.4%

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

   4. *-commutative 99.4%

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

   5. 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. Taylor expanded in u around 0 75.5%

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

   1. *-commutative 75.5%

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

   2. *-commutative 75.5%

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

   3. associate-*l* 75.7%

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

6. Simplified 75.7%

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

7. Final simplification 75.7%

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

Reproduce

herbie shell --seed 2023306 
(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))))))