Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.3%

Time: 6.5s

Alternatives: 17

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: 95.9% 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: 98.3% 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

\[-\left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot s\right) \cdot 3 \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (- (* (* (log1p (/ (- 0.25 u) 0.75)) s) 3.0)))

C

float code(float s, float u) {
	return -((log1pf(((0.25f - u) / 0.75f)) * s) * 3.0f);
}

Julia

function code(s, u)
	return Float32(-Float32(Float32(log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75))) * s) * Float32(3.0)))
end

Derivation

1. Initial program 95.9%

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

3. Applied rewrites 96.3%

   \[\leadsto -\left(\log \left(\left(u - 1\right) \cdot -1.3333333333333333\right) \cdot s\right) \cdot 3 \]
4. Step-by-step derivation

5. Applied rewrites 97.9%

   \[\leadsto -\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)\right) \cdot s\right) \cdot 3 \]
6. Step-by-step derivation

7. Applied rewrites 98.3%

   \[\leadsto -\left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot s\right) \cdot 3 \]
8. Add Preprocessing

Alternative 2: 98.2% accurate, 1.1× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

3. Applied rewrites 96.3%

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

5. Applied rewrites 95.6%

   \[\leadsto \mathsf{fma}\left(\log \left(\left|u - 1\right|\right), -3 \cdot s, \log 1.3333333333333333 \cdot \left(-3 \cdot s\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.4%

   \[\leadsto \frac{\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right) \cdot s}{-0.3333333333333333} \]
8. Step-by-step derivation

9. Applied rewrites 98.2%

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

Alternative 3: 98.2% accurate, 1.1× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

3. Applied rewrites 96.3%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 97.9%

   \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.8%

   \[\leadsto s \cdot \frac{\log \left(1 - \mathsf{fma}\left(1.3333333333333333, u, -0.3333333333333333\right)\right)}{-0.3333333333333333} \]

9. Applied rewrites 98.2%

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

Alternative 4: 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

3. Applied rewrites 96.3%

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

5. Applied rewrites 96.9%

   \[\leadsto -3 \cdot \left(\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right) \cdot s\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.3%

   \[\leadsto -3 \cdot \left(\log \left(-1.3333333333333333 \cdot \left(u - 1\right)\right) \cdot s\right) \]
8. Step-by-step derivation

9. Applied rewrites 97.9%

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

Alternative 5: 97.0% 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

\[\left(\left(\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right) \cdot -1.3333333333333333\right) \cdot s\right) \cdot 2.25 \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (*
 (*
  (*
   (log (fma -1.3333333333333333 u 1.3333333333333333))
   -1.3333333333333333)
  s)
 2.25))

C

float code(float s, float u) {
	return ((logf(fmaf(-1.3333333333333333f, u, 1.3333333333333333f)) * -1.3333333333333333f) * s) * 2.25f;
}

Julia

function code(s, u)
	return Float32(Float32(Float32(log(fma(Float32(-1.3333333333333333), u, Float32(1.3333333333333333))) * Float32(-1.3333333333333333)) * s) * Float32(2.25))
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

3. Applied rewrites 96.3%

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

5. Applied rewrites 95.6%

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

7. Applied rewrites 97.0%

   \[\leadsto \left(\left(\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right) \cdot -1.3333333333333333\right) \cdot s\right) \cdot 2.25 \]
8. Add Preprocessing

Alternative 6: 97.0% 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

\[\left(-2.25 \cdot \log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right) \cdot \left(1.3333333333333333 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (*
 (* -2.25 (log (fma -1.3333333333333333 u 1.3333333333333333)))
 (* 1.3333333333333333 s)))

C

float code(float s, float u) {
	return (-2.25f * logf(fmaf(-1.3333333333333333f, u, 1.3333333333333333f))) * (1.3333333333333333f * s);
}

Julia

function code(s, u)
	return Float32(Float32(Float32(-2.25) * log(fma(Float32(-1.3333333333333333), u, Float32(1.3333333333333333)))) * Float32(Float32(1.3333333333333333) * s))
end

Derivation

1. Initial program 95.9%

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

3. Applied rewrites 96.3%

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

5. Applied rewrites 95.8%

   \[\leadsto \log \left(\sqrt{\left|\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right|}\right) \cdot \left(s \cdot -6\right) \]
6. Step-by-step derivation

7. Applied rewrites 95.8%

   \[\leadsto -6 \cdot \left(\log \left(\sqrt{\left|\mathsf{fma}\left(1.3333333333333333, u, -1.3333333333333333\right)\right|}\right) \cdot s\right) \]

9. Applied rewrites 97.0%

   \[\leadsto \left(-2.25 \cdot \log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right) \cdot \left(1.3333333333333333 \cdot s\right) \]
10. Add Preprocessing

Alternative 7: 96.9% 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

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

FPCore

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

C

float code(float s, float u) {
	return s * fmaf(-3.0f, logf(fabsf((u - 1.0f))), -0.8630462288856506f);
}

Julia

function code(s, u)
	return Float32(s * fma(Float32(-3.0), log(abs(Float32(u - Float32(1.0)))), Float32(-0.8630462288856506)))
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

3. Applied rewrites 96.3%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 96.8%

   \[\leadsto s \cdot \mathsf{fma}\left(-3, \log \left(\left|u - 1\right|\right), \log 0.421875\right) \]

6. Evaluated real constant 96.8%

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

Alternative 8: 96.9% 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

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

FPCore

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

C

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

Julia

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

3. Applied rewrites 96.3%

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

5. Applied rewrites 96.9%

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

Alternative 9: 96.8% 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

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

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

Derivation

1. Initial program 95.9%

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

3. Applied rewrites 96.3%

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

5. Applied rewrites 96.9%

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

Alternative 10: 36.7% 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

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

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 95.9%

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

3. Applied rewrites 96.3%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 96.5%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left|u - 1\right|\right) + \log 0.421875\right) \]

6. Evaluated real constant 96.5%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left|u - 1\right|\right) + -0.8630462288856506\right) \]

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(u \cdot \left(3 + u \cdot \left(\frac{3}{2} + u\right)\right) + -0.8630462288856506\right) \]
8. Step-by-step derivation

9. Applied rewrites 36.7%

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

Alternative 11: 32.2% 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 95.9%

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

3. Applied rewrites 96.3%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 96.5%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left|u - 1\right|\right) + \log 0.421875\right) \]

6. Evaluated real constant 96.5%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left|u - 1\right|\right) + -0.8630462288856506\right) \]

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(u \cdot \left(3 + \frac{3}{2} \cdot u\right) + -0.8630462288856506\right) \]
8. Step-by-step derivation

9. Applied rewrites 32.2%

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

Alternative 12: 25.8% 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 95.9%

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

3. Applied rewrites 96.3%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 96.5%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left|u - 1\right|\right) + \log 0.421875\right) \]

6. Taylor expanded in u around 0

   \[\leadsto s \cdot \left(3 \cdot u + \log 0.421875\right) \]
7. Step-by-step derivation

8. Applied rewrites 25.8%

   \[\leadsto s \cdot \left(3 \cdot u + \log 0.421875\right) \]

9. Evaluated real constant 25.8%

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

Alternative 13: 25.8% 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

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

FPCore

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

C

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

Fortran

real(4) function code(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 3.0e0 * (((-0.28768208622932434e0) + u) * s)
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = single(3.0) * ((single(-0.28768208622932434) + u) * 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. Taylor expanded in u around 0

   \[\leadsto \left(3 \cdot s\right) \cdot \left(u + \log \frac{3}{4}\right) \]
3. Step-by-step derivation

4. Applied rewrites 25.8%

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

6. Applied rewrites 25.8%

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

7. Evaluated real constant 25.8%

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

Alternative 14: 10.5% 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 95.9%

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

2. Taylor expanded in u around 0

   \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]
3. Step-by-step derivation

4. Applied rewrites 7.4%

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

5. Evaluated real constant 7.4%

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

6. Taylor expanded in undef-var around zero

   \[\leadsto 0 \cdot \left(s \cdot -0.28768208622932434\right) \]
7. Step-by-step derivation

8. Applied rewrites 10.5%

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

Alternative 15: 7.4% 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

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

FPCore

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

C

float code(float s, float u) {
	return 3.0f * (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 = 3.0e0 * (s * (-0.28768208622932434e0))
end function

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = single(3.0) * (s * single(-0.28768208622932434));
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. Taylor expanded in u around 0

   \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]
3. Step-by-step derivation

4. Applied rewrites 7.4%

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

5. Evaluated real constant 7.4%

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

Alternative 16: 7.4% 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 95.9%

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

3. Applied rewrites 96.3%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left(u - 1\right) \cdot -1.3333333333333333\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 96.5%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left|u - 1\right|\right) + \log 0.421875\right) \]

6. Evaluated real constant 96.5%

   \[\leadsto s \cdot \left(-3 \cdot \log \left(\left|u - 1\right|\right) + -0.8630462288856506\right) \]

7. Taylor expanded in u around 0

   \[\leadsto s \cdot \frac{-14479513}{16777216} \]
8. Step-by-step derivation

9. Applied rewrites 7.4%

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

Alternative 17: 7.4% 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 95.9%

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

2. Taylor expanded in u around 0

   \[\leadsto 3 \cdot \left(s \cdot \log \frac{3}{4}\right) \]
3. Step-by-step derivation

4. Applied rewrites 7.4%

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

5. Evaluated real constant 7.4%

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

7. Applied rewrites 7.4%

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

Reproduce

herbie shell --seed 2026084 
(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))))))