Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 97.3%

Time: 53.1s

Alternatives: 9

Speedup: 1.4×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Initial Program: 95.9% accurate, 1.0× speedup

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Alternative 1: 97.3% accurate, 0.5× speedup

Math

\[s \cdot \mathsf{fma}\left(-1, \log \left(\left|\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right)\right|\right), \left(-\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right) - 0.5 \cdot \log \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 s
 (fma
  -1.0
  (log (fabs (fma -1.7777777777777777 u 1.7777777777777777)))
  (-
   (- (log (fma -1.3333333333333333 u 1.3333333333333333)))
   (* 0.5 (log (* (- 1.0 u) (- 1.0 u))))))))

C

float code(float s, float u) {
	return s * fmaf(-1.0f, logf(fabsf(fmaf(-1.7777777777777777f, u, 1.7777777777777777f))), (-logf(fmaf(-1.3333333333333333f, u, 1.3333333333333333f)) - (0.5f * logf(((1.0f - u) * (1.0f - u))))));
}

Julia

function code(s, u)
	return Float32(s * fma(Float32(-1.0), log(abs(fma(Float32(-1.7777777777777777), u, Float32(1.7777777777777777)))), Float32(Float32(-log(fma(Float32(-1.3333333333333333), u, Float32(1.3333333333333333)))) - Float32(Float32(0.5) * log(Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) - u)))))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.9%

   \[\leadsto \color{blue}{s \cdot \log \left(\left|\frac{1}{\frac{\left(\left(u - 1\right) \cdot \left(1 - u\right)\right) \cdot -1.7777777777777777}{1}} \cdot \frac{1}{\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)}\right|\right)} \]

5. Applied rewrites 97.3%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(-1, \log \left(\left|\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right)\right|\right), \left(-\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right) - 0.5 \cdot \log \left(\left(1 - u\right) \cdot \left(1 - u\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 97.3% accurate, 0.7× speedup

Math

\[s \cdot \mathsf{fma}\left(-1, \log \left(\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)\right), -\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 s
 (fma
  -1.0
  (log (* (fma -1.7777777777777777 u 1.7777777777777777) (- 1.0 u)))
  (- (log (fma -1.3333333333333333 u 1.3333333333333333))))))

C

float code(float s, float u) {
	return s * fmaf(-1.0f, logf((fmaf(-1.7777777777777777f, u, 1.7777777777777777f) * (1.0f - u))), -logf(fmaf(-1.3333333333333333f, u, 1.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(s * fma(Float32(-1.0), log(Float32(fma(Float32(-1.7777777777777777), u, Float32(1.7777777777777777)) * Float32(Float32(1.0) - u))), Float32(-log(fma(Float32(-1.3333333333333333), u, Float32(1.3333333333333333))))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.9%

   \[\leadsto \color{blue}{s \cdot \log \left(\left|\frac{1}{\frac{\left(\left(u - 1\right) \cdot \left(1 - u\right)\right) \cdot -1.7777777777777777}{1}} \cdot \frac{1}{\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)}\right|\right)} \]

5. Applied rewrites 97.3%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(-1, \log \left(\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)\right), -\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right)} \]
6. Add Preprocessing

Alternative 3: 97.2% accurate, 0.7× speedup

Math

\[s \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)}{\frac{1}{\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)}}}\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 s
 (log
  (/
   1.0
   (/
    (* (fma -1.7777777777777777 u 1.7777777777777777) (- 1.0 u))
    (/ 1.0 (fma -1.3333333333333333 u 1.3333333333333333)))))))

C

float code(float s, float u) {
	return s * logf((1.0f / ((fmaf(-1.7777777777777777f, u, 1.7777777777777777f) * (1.0f - u)) / (1.0f / fmaf(-1.3333333333333333f, u, 1.3333333333333333f)))));
}

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(fma(Float32(-1.7777777777777777), u, Float32(1.7777777777777777)) * Float32(Float32(1.0) - u)) / Float32(Float32(1.0) / fma(Float32(-1.3333333333333333), u, Float32(1.3333333333333333)))))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.9%

   \[\leadsto \color{blue}{s \cdot \log \left(\left|\frac{1}{\frac{\left(\left(u - 1\right) \cdot \left(1 - u\right)\right) \cdot -1.7777777777777777}{1}} \cdot \frac{1}{\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)}\right|\right)} \]

5. Applied rewrites 97.2%

   \[\leadsto s \cdot \log \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)}{\frac{1}{\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)}}}\right)} \]
6. Add Preprocessing

Alternative 4: 97.1% accurate, 1.0× speedup

Math

\[\left(3 \cdot s\right) \cdot \left(-0.5 \cdot \log \left(\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (* 3.0 s)
 (*
  -0.5
  (log (* (fma -1.7777777777777777 u 1.7777777777777777) (- 1.0 u))))))

C

float code(float s, float u) {
	return (3.0f * s) * (-0.5f * logf((fmaf(-1.7777777777777777f, u, 1.7777777777777777f) * (1.0f - u))));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(Float32(-0.5) * log(Float32(fma(Float32(-1.7777777777777777), u, Float32(1.7777777777777777)) * Float32(Float32(1.0) - u)))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(0.5 \cdot \log \left(\left|\frac{1}{\frac{\left(\left(u - 1\right) \cdot \left(1 - u\right)\right) \cdot -1.7777777777777777}{1}}\right|\right)\right)} \]

5. Applied rewrites 97.1%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(-0.5 \cdot \log \left(\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)\right)\right)} \]
6. Add Preprocessing

Alternative 5: 97.1% accurate, 1.1× speedup

Math

\[s \cdot \left(-1.5 \cdot \log \left(\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 s
 (*
  -1.5
  (log (* (fma -1.7777777777777777 u 1.7777777777777777) (- 1.0 u))))))

C

float code(float s, float u) {
	return s * (-1.5f * logf((fmaf(-1.7777777777777777f, u, 1.7777777777777777f) * (1.0f - u))));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-1.5) * log(Float32(fma(Float32(-1.7777777777777777), u, Float32(1.7777777777777777)) * Float32(Float32(1.0) - u)))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.9%

   \[\leadsto \color{blue}{s \cdot \log \left(\left|\frac{1}{\frac{\left(\left(u - 1\right) \cdot \left(1 - u\right)\right) \cdot -1.7777777777777777}{1}} \cdot \frac{1}{\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)}\right|\right)} \]

5. Applied rewrites 97.1%

   \[\leadsto s \cdot \color{blue}{\left(-1.5 \cdot \log \left(\mathsf{fma}\left(-1.7777777777777777, u, 1.7777777777777777\right) \cdot \left(1 - u\right)\right)\right)} \]
6. Add Preprocessing

Alternative 6: 96.8% accurate, 1.4× speedup

Math

\[\left(\log \left(\left|\mathsf{fma}\left(1.3333333333333333, u, -1.3333333333333333\right)\right|\right) \cdot s\right) \cdot -3 \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (* (log (fabs (fma 1.3333333333333333 u -1.3333333333333333))) s)
 -3.0))

C

float code(float s, float u) {
	return (logf(fabsf(fmaf(1.3333333333333333f, u, -1.3333333333333333f))) * s) * -3.0f;
}

Julia

function code(s, u)
	return Float32(Float32(log(abs(fma(Float32(1.3333333333333333), u, Float32(-1.3333333333333333)))) * s) * Float32(-3.0))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\left(\log \left(\left|\mathsf{fma}\left(1.3333333333333333, u, -1.3333333333333333\right)\right|\right) \cdot s\right) \cdot -3} \]
4. Add Preprocessing

Alternative 7: 96.8% accurate, 1.4× speedup

Math

\[\left(-3 \cdot s\right) \cdot \log \left(\left|\mathsf{fma}\left(1.3333333333333333, u, -1.3333333333333333\right)\right|\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (* -3.0 s)
 (log (fabs (fma 1.3333333333333333 u -1.3333333333333333)))))

C

float code(float s, float u) {
	return (-3.0f * s) * logf(fabsf(fmaf(1.3333333333333333f, u, -1.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(-3.0) * s) * log(abs(fma(Float32(1.3333333333333333), u, Float32(-1.3333333333333333)))))
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \log \left(\left|\mathsf{fma}\left(1.3333333333333333, u, -1.3333333333333333\right)\right|\right)} \]
4. Add Preprocessing

Alternative 8: 96.8% accurate, 1.4× speedup

Math

\[\left(-3 \cdot \log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right) \cdot s \]

FPCore

(FPCore (s u)
  :precision binary32
  (* (* -3.0 (log (fma -1.3333333333333333 u 1.3333333333333333))) s))

C

float code(float s, float u) {
	return (-3.0f * logf(fmaf(-1.3333333333333333f, u, 1.3333333333333333f))) * s;
}

Julia

function code(s, u)
	return Float32(Float32(Float32(-3.0) * log(fma(Float32(-1.3333333333333333), u, Float32(1.3333333333333333)))) * s)
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

3. Applied rewrites 96.9%

   \[\leadsto \color{blue}{s \cdot \log \left(\left|\frac{1}{\frac{\left(\left(u - 1\right) \cdot \left(1 - u\right)\right) \cdot -1.7777777777777777}{1}} \cdot \frac{1}{\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)}\right|\right)} \]

5. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\left(-3 \cdot \log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right) \cdot s} \]
6. Add Preprocessing

Alternative 9: 27.8% accurate, 7.2× speedup

Math

\[s + s \]

FPCore

(FPCore (s u)
  :precision binary32
  (+ s s))

C

float code(float s, float u) {
	return s + s;
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s + s
end function

Julia

function code(s, u)
	return Float32(s + s)
end

MATLAB

function tmp = code(s, u)
	tmp = s + s;
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   3. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   4. frac-2neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)}{\mathsf{neg}\left(\frac{3}{4}\right)}}}\right) \]

   5. frac-2neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}}\right) \]

   6. remove-double-neg N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}\right) \]

   7. /-rgt-identity N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{1}}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\frac{\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)}{1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}\right) \]

   9. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\frac{\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}}{1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}\right) \]

   10. associate--r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\frac{\color{blue}{\left(\frac{3}{4} - u\right) + \frac{1}{4}}}{1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}\right) \]

   11. div-add N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{\frac{\frac{3}{4} - u}{1} + \frac{\frac{1}{4}}{1}}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\frac{\frac{3}{4} - u}{1} + \color{blue}{\frac{1}{4}}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)}}\right) \]

   13. remove-double-neg N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\frac{\frac{3}{4} - u}{1} + \frac{1}{4}}{\color{blue}{\frac{3}{4}}}}\right) \]

   14. div-add N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\frac{\frac{3}{4} - u}{1}}{\frac{3}{4}} + \frac{\frac{1}{4}}{\frac{3}{4}}}}\right) \]

3. Applied rewrites 96.0%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\mathsf{fma}\left(0.75 \cdot \left(u - 0.75\right), -1.7777777777777777, 0.3333333333333333\right)}}\right) \]

5. Applied rewrites 27.8%

   \[\leadsto \color{blue}{s + s} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))