Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.7% → 99.4%

Time: 8.2s

Alternatives: 11

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: 61.7% 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 60.9%

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

   1. log-rec 63.4%

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

   2. distribute-rgt-neg-out 63.4%

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

   3. distribute-lft-neg-out 63.4%

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

   4. cancel-sign-sub-inv 63.4%

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

   5. log1p-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 93.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 4 + u \cdot \left(s \cdot -8 + \left(s \cdot 16 + u \cdot \left(s \cdot 21.333333333333332 + u \cdot \left(s \cdot -64 + s \cdot 128\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  u
  (+
   (* s 4.0)
   (*
    u
    (+
     (* s -8.0)
     (+
      (* s 16.0)
      (*
       u
       (+ (* s 21.333333333333332) (* u (+ (* s -64.0) (* s 128.0)))))))))))

C

float code(float s, float u) {
	return u * ((s * 4.0f) + (u * ((s * -8.0f) + ((s * 16.0f) + (u * ((s * 21.333333333333332f) + (u * ((s * -64.0f) + (s * 128.0f)))))))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * 4.0e0) + (u * ((s * (-8.0e0)) + ((s * 16.0e0) + (u * ((s * 21.333333333333332e0) + (u * ((s * (-64.0e0)) + (s * 128.0e0)))))))))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(s * Float32(4.0)) + Float32(u * Float32(Float32(s * Float32(-8.0)) + Float32(Float32(s * Float32(16.0)) + Float32(u * Float32(Float32(s * Float32(21.333333333333332)) + Float32(u * Float32(Float32(s * Float32(-64.0)) + Float32(s * Float32(128.0)))))))))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * single(4.0)) + (u * ((s * single(-8.0)) + ((s * single(16.0)) + (u * ((s * single(21.333333333333332)) + (u * ((s * single(-64.0)) + (s * single(128.0))))))))));
end

Derivation

1. Initial program 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(1 + -4 \cdot u\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. neg-log 60.9%

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

   2. flip-+ 60.4%

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

   3. associate-/r/ 60.0%

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

   4. log-prod 60.0%

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

   5. metadata-eval 60.0%

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

   6. pow2 60.0%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - \color{blue}{{\left(-4 \cdot u\right)}^{2}}}\right) + \log \left(1 - -4 \cdot u\right)\right) \]

   7. *-commutative 60.0%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {\color{blue}{\left(u \cdot -4\right)}}^{2}}\right) + \log \left(1 - -4 \cdot u\right)\right) \]

   8. unpow-prod-down 60.0%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - \color{blue}{{u}^{2} \cdot {-4}^{2}}}\right) + \log \left(1 - -4 \cdot u\right)\right) \]

   9. metadata-eval 60.0%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{2} \cdot \color{blue}{16}}\right) + \log \left(1 - -4 \cdot u\right)\right) \]

   10. cancel-sign-sub-inv 60.0%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{2} \cdot 16}\right) + \log \color{blue}{\left(1 + \left(--4\right) \cdot u\right)}\right) \]

   11. log1p-define 87.0%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{2} \cdot 16}\right) + \color{blue}{\mathsf{log1p}\left(\left(--4\right) \cdot u\right)}\right) \]

   12. metadata-eval 87.0%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{2} \cdot 16}\right) + \mathsf{log1p}\left(\color{blue}{4} \cdot u\right)\right) \]

6. Applied egg-rr 87.0%

   \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - {u}^{2} \cdot 16}\right) + \mathsf{log1p}\left(4 \cdot u\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 87.0%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) + \log \left(\frac{1}{1 - {u}^{2} \cdot 16}\right)\right)} \]

   2. log-rec 88.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) + \color{blue}{\left(-\log \left(1 - {u}^{2} \cdot 16\right)\right)}\right) \]

   3. sub-neg 88.4%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) + \left(-\log \color{blue}{\left(1 + \left(-{u}^{2} \cdot 16\right)\right)}\right)\right) \]

   4. log1p-undefine 99.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) + \left(-\color{blue}{\mathsf{log1p}\left(-{u}^{2} \cdot 16\right)}\right)\right) \]

   5. unsub-neg 99.0%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(-{u}^{2} \cdot 16\right)\right)} \]

   6. *-commutative 99.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{u \cdot 4}\right) - \mathsf{log1p}\left(-{u}^{2} \cdot 16\right)\right) \]

   7. distribute-rgt-neg-in 99.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(u \cdot 4\right) - \mathsf{log1p}\left(\color{blue}{{u}^{2} \cdot \left(-16\right)}\right)\right) \]

   8. metadata-eval 99.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(u \cdot 4\right) - \mathsf{log1p}\left({u}^{2} \cdot \color{blue}{-16}\right)\right) \]

8. Simplified 99.0%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(u \cdot 4\right) - \mathsf{log1p}\left({u}^{2} \cdot -16\right)\right)} \]
9. Step-by-step derivation

   1. sub-neg 99.0%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(u \cdot 4\right) + \left(-\mathsf{log1p}\left({u}^{2} \cdot -16\right)\right)\right)} \]

   2. distribute-rgt-in 99.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(u \cdot 4\right) \cdot s + \left(-\mathsf{log1p}\left({u}^{2} \cdot -16\right)\right) \cdot s} \]

10. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{s \cdot \mathsf{log1p}\left(u \cdot 4\right) + \left(-\mathsf{log1p}\left({u}^{2} \cdot -16\right)\right) \cdot s} \]

11. Taylor expanded in u around 0 94.7%

    \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + u \cdot \left(-8 \cdot s + \left(16 \cdot s + u \cdot \left(21.333333333333332 \cdot s + u \cdot \left(-64 \cdot s + 128 \cdot s\right)\right)\right)\right)\right)} \]

12. Final simplification 94.7%

    \[\leadsto u \cdot \left(s \cdot 4 + u \cdot \left(s \cdot -8 + \left(s \cdot 16 + u \cdot \left(s \cdot 21.333333333333332 + u \cdot \left(s \cdot -64 + s \cdot 128\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 93.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 4 + u \cdot \left(s \cdot 8 + u \cdot \left(s \cdot 21.333333333333332 + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  u
  (+
   (* s 4.0)
   (* u (+ (* s 8.0) (* u (+ (* s 21.333333333333332) (* 64.0 (* s u)))))))))

C

float code(float s, float u) {
	return u * ((s * 4.0f) + (u * ((s * 8.0f) + (u * ((s * 21.333333333333332f) + (64.0f * (s * u)))))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * 4.0e0) + (u * ((s * 8.0e0) + (u * ((s * 21.333333333333332e0) + (64.0e0 * (s * u)))))))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(s * Float32(4.0)) + Float32(u * Float32(Float32(s * Float32(8.0)) + Float32(u * Float32(Float32(s * Float32(21.333333333333332)) + Float32(Float32(64.0) * Float32(s * u))))))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * single(4.0)) + (u * ((s * single(8.0)) + (u * ((s * single(21.333333333333332)) + (single(64.0) * (s * u)))))));
end

Derivation

1. Initial program 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 94.6%

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

6. Final simplification 94.6%

   \[\leadsto u \cdot \left(s \cdot 4 + u \cdot \left(s \cdot 8 + u \cdot \left(s \cdot 21.333333333333332 + 64 \cdot \left(s \cdot u\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 93.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (+ 4.0 (* u (+ 8.0 (* u (+ 21.333333333333332 (* u 64.0)))))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * (8.0f + (u * (21.333333333333332f + (u * 64.0f)))))));
}

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 + (u * (21.333333333333332e0 + (u * 64.0e0)))))))
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(Float32(21.333333333333332) + Float32(u * Float32(64.0)))))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * (single(8.0) + (u * (single(21.333333333333332) + (u * single(64.0))))))));
end

Derivation

1. Initial program 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 94.2%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + 64 \cdot u\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 94.2%

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

7. Simplified 94.2%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot \left(21.333333333333332 + u \cdot 64\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 5: 91.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 92.8%

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

6. Taylor expanded in s around 0 92.8%

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

   1. *-commutative 92.8%

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

8. Simplified 92.8%

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

9. Final simplification 92.8%

   \[\leadsto u \cdot \left(s \cdot 4 + u \cdot \left(s \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 90.9% 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 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 92.4%

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

   1. *-commutative 92.4%

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

7. Simplified 92.4%

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

Alternative 7: 86.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 88.6%

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

6. Final simplification 88.6%

   \[\leadsto u \cdot \left(s \cdot 4 + 8 \cdot \left(s \cdot u\right)\right) \]
7. Add Preprocessing

Alternative 8: 86.7% 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 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 88.3%

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

   1. *-commutative 88.3%

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

7. Simplified 88.3%

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

   1. +-commutative 88.3%

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

   2. distribute-lft-in 88.5%

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

9. Applied egg-rr 88.5%

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

Alternative 9: 86.5% 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 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 88.3%

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

   1. *-commutative 88.3%

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

7. Simplified 88.3%

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

Alternative 10: 73.5% 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 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 74.7%

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

   1. *-commutative 74.7%

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

   2. *-commutative 74.7%

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

   3. associate-*r* 75.0%

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

7. Simplified 75.0%

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

Alternative 11: 73.3% 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 60.9%

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

   1. log-rec 63.4%

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

   2. cancel-sign-sub-inv 63.4%

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

   3. metadata-eval 63.4%

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

3. Simplified 63.4%

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

5. Taylor expanded in u around 0 74.7%

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

Reproduce

herbie shell --seed 2024157 
(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))))))