Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 9.5s

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: 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 61.4%

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

   1. log-rec 63.6%

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

   2. distribute-rgt-neg-out 63.6%

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

   3. distribute-lft-neg-in 63.6%

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

   4. cancel-sign-sub-inv 63.6%

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

   5. log1p-def 99.3%

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

   6. *-commutative 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 86.8% accurate, 8.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (s * (u * 4.0f)) + (s * (u * (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)) + (s * (u * (u * 8.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

2. Taylor expanded in u around 0 93.6%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + \left(8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)\right)} \]

3. Taylor expanded in u around 0 86.7%

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

   1. *-commutative 86.7%

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

   2. unpow2 86.7%

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

   3. associate-*l* 86.7%

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

5. Simplified 86.7%

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

   1. distribute-lft-in 86.7%

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

   2. *-commutative 86.7%

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

7. Applied egg-rr 86.7%

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

8. Final simplification 86.7%

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

Alternative 3: 86.8% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

2. Taylor expanded in u around 0 93.6%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + \left(8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)\right)} \]

3. Taylor expanded in u around 0 86.7%

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

   1. *-commutative 86.7%

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

   2. unpow2 86.7%

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

   3. associate-*l* 86.7%

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

5. Simplified 86.7%

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

6. Final simplification 86.7%

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

Alternative 4: 86.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.4%

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

2. Taylor expanded in u around 0 93.6%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + \left(8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)\right)} \]

3. Taylor expanded in u around 0 86.7%

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

   1. *-commutative 86.7%

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

   2. unpow2 86.7%

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

   3. associate-*l* 86.7%

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

5. Simplified 86.7%

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

   1. distribute-lft-in 86.7%

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

   2. *-commutative 86.7%

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

7. Applied egg-rr 86.7%

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

   1. associate-*r* 86.6%

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

   2. *-commutative 86.6%

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

   3. associate-*r* 86.6%

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

   4. *-commutative 86.6%

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

   5. distribute-lft-in 86.4%

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

   6. associate-*l* 86.5%

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

9. Applied egg-rr 86.5%

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

10. Final simplification 86.5%

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

Alternative 5: 73.8% 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 61.4%

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

2. Taylor expanded in u around 0 73.1%

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

   1. *-commutative 73.1%

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

4. Simplified 73.1%

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

5. Final simplification 73.1%

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

Alternative 6: 74.0% 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 61.4%

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

2. Taylor expanded in u around 0 73.1%

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

   1. associate-*r* 73.3%

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

4. Simplified 73.3%

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

5. Final simplification 73.3%

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

Alternative 7: 17.0% 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 61.4%

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

3. Applied egg-rr 16.6%

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

   1. +-inverses 16.6%

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

5. Simplified 16.6%

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

6. Final simplification 16.6%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023271 
(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))))))