Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.4%

Time: 11.3s

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: 61.3% 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 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 99.4%

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

   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. Add Preprocessing

5. Final simplification 99.4%

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

Alternative 2: 93.4% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

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

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * single(4.0)) + (u * ((s * single(8.0)) + (u * (s * (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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 95.0%

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

6. Taylor expanded in s around 0 95.0%

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

   1. *-commutative 95.0%

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

8. Simplified 95.0%

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

9. Final simplification 95.0%

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

Alternative 3: 91.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return u * ((s * 4.0f) + (u * ((s * 8.0f) + (21.333333333333332f * (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) + (21.333333333333332e0 * (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(Float32(21.333333333333332) * Float32(s * u))))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * single(4.0)) + (u * ((s * single(8.0)) + (single(21.333333333333332) * (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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 93.1%

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

6. Final simplification 93.1%

   \[\leadsto u \cdot \left(s \cdot 4 + u \cdot \left(s \cdot 8 + 21.333333333333332 \cdot \left(s \cdot u\right)\right)\right) \]
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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 94.6%

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

6. Final simplification 94.6%

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

Alternative 5: 91.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * single(4.0)) + (s * (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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 93.1%

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

6. Taylor expanded in s around 0 93.1%

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

   1. *-commutative 93.1%

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

8. Simplified 93.1%

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

9. Final simplification 93.1%

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

Alternative 6: 91.1% 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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 93.1%

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

6. Taylor expanded in s around 0 92.8%

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

7. Final simplification 92.8%

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

Alternative 7: 91.1% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = u * (s * (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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 93.1%

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

6. Taylor expanded in s around 0 92.8%

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

   1. *-commutative 92.8%

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

8. Simplified 92.8%

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

9. Final simplification 92.8%

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

Alternative 8: 87.0% 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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 88.7%

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

6. Final simplification 88.7%

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

Alternative 9: 86.8% 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. 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 88.7%

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

   1. *-commutative 88.7%

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

   2. associate-*r* 88.7%

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

   3. distribute-rgt-out 88.5%

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

   4. *-commutative 88.5%

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

7. Simplified 88.5%

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

8. Taylor expanded in s around 0 88.5%

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

9. Final simplification 88.5%

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

Alternative 10: 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 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 75.7%

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

6. Final simplification 75.7%

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

Alternative 11: 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 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 99.4%

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

   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. Add Preprocessing

5. Taylor expanded in u around 0 75.7%

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

   1. associate-*r* 76.0%

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

   2. *-commutative 76.0%

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

7. Simplified 76.0%

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

8. Final simplification 76.0%

   \[\leadsto u \cdot \left(s \cdot 4\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(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))))))