Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.6% → 99.4%

Time: 10.5s

Alternatives: 9

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. cancel-sign-sub-inv 61.0%

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

   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: 91.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (((u * (u * 21.333333333333332e0)) + (u * 8.0e0)) * (s * u)) + (u * (s * 4.0e0))
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (((u * (u * single(21.333333333333332))) + (u * single(8.0))) * (s * u)) + (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 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. *-commutative 92.2%

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

   6. cube-mult 92.2%

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

   7. unpow2 92.2%

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

   8. associate-*l* 92.2%

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

   9. distribute-lft-out 92.2%

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

   10. distribute-lft-out 91.9%

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

   11. unpow2 91.9%

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

   12. associate-*l* 91.9%

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

   13. 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. Step-by-step derivation

   1. associate-*r* 91.7%

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

   2. *-commutative 91.7%

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

   3. +-commutative 91.7%

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

   4. distribute-rgt-in 92.0%

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

   5. +-commutative 92.0%

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

   6. *-commutative 92.0%

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

   7. fma-def 92.0%

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

   8. *-commutative 92.0%

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

   9. associate-*l* 92.2%

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

6. Applied egg-rr 92.2%

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

   1. fma-udef 92.2%

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

   2. distribute-rgt-in 92.2%

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

   3. *-commutative 92.2%

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

   4. *-commutative 92.2%

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

8. Applied egg-rr 92.2%

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

9. Final simplification 92.2%

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

Alternative 3: 91.2% accurate, 8.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * ((u * 21.333333333333332f) + 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 * ((u * 21.333333333333332e0) + 8.0e0))))
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * ((u * single(21.333333333333332)) + 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 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. *-commutative 92.2%

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

   6. cube-mult 92.2%

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

   7. unpow2 92.2%

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

   8. associate-*l* 92.2%

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

   9. distribute-lft-out 92.2%

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

   10. distribute-lft-out 91.9%

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

   11. unpow2 91.9%

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

   12. associate-*l* 91.9%

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

   13. 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%

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

Alternative 4: 87.3% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = u * ((u * (s * single(8.0))) + (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 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%

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

   2. associate-*r* 88.3%

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

   3. unpow2 88.3%

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

   4. associate-*r* 88.3%

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

   5. associate-*r* 88.5%

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

   6. distribute-rgt-out 88.5%

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

   7. *-commutative 88.5%

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

   8. associate-*l* 88.5%

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

   9. *-commutative 88.5%

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

   10. *-commutative 88.5%

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

   11. distribute-lft-out 88.3%

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

4. Simplified 88.3%

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

   1. distribute-rgt-in 88.5%

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

   2. associate-*l* 88.5%

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

6. Applied egg-rr 88.5%

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

7. Final simplification 88.5%

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

Alternative 5: 87.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * (u * ((u * single(8.0)) + 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 88.4%

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

   1. unpow2 88.4%

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

   2. associate-*r* 88.4%

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

   3. distribute-rgt-out 88.3%

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

   4. *-commutative 88.3%

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

4. Simplified 88.3%

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

5. Final simplification 88.3%

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

Alternative 6: 87.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = u * (s * ((u * single(8.0)) + 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 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%

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

   2. associate-*r* 88.3%

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

   3. unpow2 88.3%

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

   4. associate-*r* 88.3%

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

   5. associate-*r* 88.5%

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

   6. distribute-rgt-out 88.5%

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

   7. *-commutative 88.5%

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

   8. associate-*l* 88.5%

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

   9. *-commutative 88.5%

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

   10. *-commutative 88.5%

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

   11. distribute-lft-out 88.3%

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

4. Simplified 88.3%

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

5. Final simplification 88.3%

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

Alternative 7: 74.2% accurate, 21.8× speedup

Math

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

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

MATLAB

function tmp = code(s, u)
	tmp = (s * u) * 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. *-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%

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

Alternative 8: 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 9: 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))))))