Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 1.4

Time: 37.8s

Precision: binary32

Cost: 10720

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\\ \left(3 \cdot s\right) \cdot \left(\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))
   (* (* 3.0 s) (* (/ 1.0 t_0) (* t_0 t_0)))))

C

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

↓

float code(float s, float u) {
	float t_0 = logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
	return (3.0f * s) * ((1.0f / t_0) * (t_0 * t_0));
}

Fortran

real(4) function code(s, u)
    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

↓

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    t_0 = log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
    code = (3.0e0 * s) * ((1.0e0 / t_0) * (t_0 * t_0))
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

↓

function code(s, u)
	t_0 = log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75)))))
	return Float32(Float32(Float32(3.0) * s) * Float32(Float32(Float32(1.0) / t_0) * Float32(t_0 * t_0)))
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

↓

function tmp = code(s, u)
	t_0 = log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
	tmp = (single(3.0) * s) * ((single(1.0) / t_0) * (t_0 * t_0));
end

Derivation

1. Initial program 1.3

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

2. Applied egg-rr 1.4

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

3. Final simplification 1.4

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

Alternatives

Alternative 1
Error	1.4
Cost	3616

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

Alternative 2
Error	1.3
Cost	3616

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

Alternative 3
Error	1.3
Cost	3616

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

Alternative 4
Error	1.4
Cost	3552

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

Alternative 5
Error	1.4
Cost	3552

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

Alternative 6
Error	22.4
Cost	160

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

Reproduce

herbie shell --seed 2023077 
(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))))))