Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 97.9%

Time: 44.5s

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
  :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))))))

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: 95.8% accurate, 1.0× speedup

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

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
  :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))))))

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: 97.9% accurate, 1.2× speedup

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

Math

\[s \cdot \left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* s (* -3.0 (log1p (fma u -1.3333333333333333 0.3333333333333333)))))

C

float code(float s, float u) {
	return s * (-3.0f * log1pf(fmaf(u, -1.3333333333333333f, 0.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(fma(u, Float32(-1.3333333333333333), Float32(0.3333333333333333)))))
end

Derivation

1. Initial program 95.8%

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

3. Applied rewrites 96.2%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]

5. Applied rewrites 95.4%

   \[\leadsto s \cdot \left(-3 \cdot \left(\log 1.3333333333333333 + \log \left(\left|1 - u\right|\right)\right)\right) \]

7. Applied rewrites 95.6%

   \[\leadsto s \cdot \left(-3 \cdot \left(\log 1.3333333333333333 + \log \left(\left|1 - 2.6666666666666665 \cdot \left(0.375 \cdot u\right)\right|\right)\right)\right) \]

9. Applied rewrites 97.9%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right)\right) \]
10. Add Preprocessing

Alternative 2: 96.7% accurate, 1.4× speedup

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

Math

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

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* -3.0 (* (log (fma u -1.3333333333333333 1.3333333333333333)) s)))

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

3. Applied rewrites 96.2%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]

5. Applied rewrites 96.7%

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

Alternative 3: 36.6% accurate, 1.4× speedup

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

Math

\[s \cdot \left(u \cdot \left(3 + u \cdot \left(1.5 + u\right)\right) - 0.8630462288856506\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* s (- (* u (+ 3.0 (* u (+ 1.5 u)))) 0.8630462288856506)))

C

float code(float s, float u) {
	return s * ((u * (3.0f + (u * (1.5f + u)))) - 0.8630462288856506f);
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((u * (3.0e0 + (u * (1.5e0 + u)))) - 0.8630462288856506e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(Float32(3.0) + Float32(u * Float32(Float32(1.5) + u)))) - Float32(0.8630462288856506)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * (single(3.0) + (u * (single(1.5) + u)))) - single(0.8630462288856506));
end

Derivation

1. Initial program 95.8%

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

3. Applied rewrites 96.2%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]

5. Applied rewrites 95.5%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -3 \cdot \log 1.3333333333333333\right) \]

6. Evaluated real constant 96.7%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -0.8630462288856506\right) \]

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(u \cdot \left(3 + u \cdot \left(\frac{3}{2} + u\right)\right) - \frac{14479513}{16777216}\right) \]

9. Applied rewrites 36.6%

   \[\leadsto s \cdot \left(u \cdot \left(3 + u \cdot \left(1.5 + u\right)\right) - 0.8630462288856506\right) \]
10. Add Preprocessing

Alternative 4: 32.0% accurate, 1.7× speedup

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

Math

\[s \cdot \left(u \cdot \left(3 + 1.5 \cdot u\right) - 0.8630462288856506\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* s (- (* u (+ 3.0 (* 1.5 u))) 0.8630462288856506)))

C

float code(float s, float u) {
	return s * ((u * (3.0f + (1.5f * u))) - 0.8630462288856506f);
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((u * (3.0e0 + (1.5e0 * u))) - 0.8630462288856506e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(Float32(3.0) + Float32(Float32(1.5) * u))) - Float32(0.8630462288856506)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * (single(3.0) + (single(1.5) * u))) - single(0.8630462288856506));
end

Derivation

1. Initial program 95.8%

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

3. Applied rewrites 96.2%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]

5. Applied rewrites 95.5%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -3 \cdot \log 1.3333333333333333\right) \]

6. Evaluated real constant 96.7%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -0.8630462288856506\right) \]

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(u \cdot \left(3 + \frac{3}{2} \cdot u\right) - \frac{14479513}{16777216}\right) \]

9. Applied rewrites 32.0%

   \[\leadsto s \cdot \left(u \cdot \left(3 + 1.5 \cdot u\right) - 0.8630462288856506\right) \]
10. Add Preprocessing

Alternative 5: 25.6% accurate, 2.7× speedup

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

Math

\[s \cdot \left(3 \cdot u - 0.8630462288856506\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* s (- (* 3.0 u) 0.8630462288856506)))

C

float code(float s, float u) {
	return s * ((3.0f * u) - 0.8630462288856506f);
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((3.0e0 * u) - 0.8630462288856506e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(3.0) * u) - Float32(0.8630462288856506)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((single(3.0) * u) - single(0.8630462288856506));
end

Derivation

1. Initial program 95.8%

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

3. Applied rewrites 96.2%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]

5. Applied rewrites 95.5%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -3 \cdot \log 1.3333333333333333\right) \]

6. Evaluated real constant 96.7%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -0.8630462288856506\right) \]

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(3 \cdot u - \frac{14479513}{16777216}\right) \]

9. Applied rewrites 25.6%

   \[\leadsto s \cdot \left(3 \cdot u - 0.8630462288856506\right) \]
10. Add Preprocessing

Alternative 6: 25.6% accurate, 2.7× speedup

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

Math

\[3 \cdot \left(s \cdot \left(u + -0.28768208622932434\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* 3.0 (* s (+ u -0.28768208622932434))))

C

float code(float s, float u) {
	return 3.0f * (s * (u + -0.28768208622932434f));
}

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 * (u + (-0.28768208622932434e0)))
end function

Julia

function code(s, u)
	return Float32(Float32(3.0) * Float32(s * Float32(u + Float32(-0.28768208622932434))))
end

MATLAB

function tmp = code(s, u)
	tmp = single(3.0) * (s * (u + single(-0.28768208622932434)));
end

Derivation

1. Initial program 95.8%

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

2. Taylor expanded in s around 0

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

4. Applied rewrites 95.6%

   \[\leadsto 3 \cdot \left(s \cdot \log \left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)\right) \]

5. Taylor expanded in u around 0

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

7. Applied rewrites 25.6%

   \[\leadsto 3 \cdot \left(s \cdot \left(u + \log 0.75\right)\right) \]

8. Evaluated real constant 25.6%

   \[\leadsto 3 \cdot \left(s \cdot \left(u + -0.28768208622932434\right)\right) \]
9. Add Preprocessing

Alternative 7: 10.6% accurate, 3.7× speedup

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

Math

\[0 \cdot \left(s \cdot -0.28768208622932434\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* 0.0 (* s -0.28768208622932434)))

C

float code(float s, float u) {
	return 0.0f * (s * -0.28768208622932434f);
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 0.0e0 * (s * (-0.28768208622932434e0))
end function

Julia

function code(s, u)
	return Float32(Float32(0.0) * Float32(s * Float32(-0.28768208622932434)))
end

MATLAB

function tmp = code(s, u)
	tmp = single(0.0) * (s * single(-0.28768208622932434));
end

Derivation

1. Initial program 95.8%

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

2. Taylor expanded in u around 0

   \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]

4. Applied rewrites 7.5%

   \[\leadsto 3 \cdot \left(s \cdot \log 0.75\right) \]

5. Evaluated real constant 7.5%

   \[\leadsto 3 \cdot \left(s \cdot -0.28768208622932434\right) \]

6. Taylor expanded in undef-var around zero

   \[\leadsto 0 \cdot \left(s \cdot -0.28768208622932434\right) \]

8. Applied rewrites 10.6%

   \[\leadsto 0 \cdot \left(s \cdot -0.28768208622932434\right) \]
9. Add Preprocessing

Alternative 8: 7.5% accurate, 6.4× speedup

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

Math

\[s \cdot -0.8630462288856506 \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* s -0.8630462288856506))

C

float code(float s, float u) {
	return s * -0.8630462288856506f;
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (-0.8630462288856506e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(-0.8630462288856506))
end

MATLAB

function tmp = code(s, u)
	tmp = s * single(-0.8630462288856506);
end

Derivation

1. Initial program 95.8%

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

3. Applied rewrites 96.2%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]

5. Applied rewrites 95.5%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -3 \cdot \log 1.3333333333333333\right) \]

6. Evaluated real constant 96.7%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|1 - u\right|\right), -0.8630462288856506\right) \]

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{-14479513}{16777216} \]

9. Applied rewrites 7.5%

   \[\leadsto s \cdot -0.8630462288856506 \]
10. Add Preprocessing

Alternative 9: 7.5% accurate, 6.4× speedup

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

Math

\[-0.863046258687973 \cdot s \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* -0.863046258687973 s))

C

float code(float s, float u) {
	return -0.863046258687973f * s;
}

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (-0.863046258687973e0) * s
end function

Julia

function code(s, u)
	return Float32(Float32(-0.863046258687973) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = single(-0.863046258687973) * s;
end

Derivation

1. Initial program 95.8%

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

2. Taylor expanded in u around 0

   \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]

4. Applied rewrites 7.5%

   \[\leadsto 3 \cdot \left(s \cdot \log 0.75\right) \]

5. Evaluated real constant 7.5%

   \[\leadsto 3 \cdot \left(s \cdot -0.28768208622932434\right) \]

7. Applied rewrites 7.5%

   \[\leadsto -0.863046258687973 \cdot s \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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))))))