Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 1.3

Time: 15.7s

Precision: binary32

Cost: 3616

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

Math

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

↓

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

FPCore

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

↓

(FPCore (s u)
 :precision binary32
 (* s (* 3.0 (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))));
}

↓

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

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
    code = s * (3.0e0 * 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

↓

function code(s, u)
	return Float32(s * Float32(Float32(3.0) * 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

↓

function tmp = code(s, u)
	tmp = s * (single(3.0) * log((single(1.0) / (single(1.0) - ((u + single(-0.25)) / single(0.75))))));
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. Simplified 1.3

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

   [Start] 1.3	\[ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 1.3	\[ \color{blue}{\log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \cdot \left(3 \cdot s\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.3	\[ \color{blue}{3 \cdot \left(\log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \cdot s\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 1.3	\[ 3 \cdot \color{blue}{\left(s \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]

3. Applied egg-rr 1.4

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

4. Applied egg-rr 1.3

   \[\leadsto \color{blue}{\log \left(\frac{1}{1 - \frac{u + -0.25}{0.75}}\right) \cdot \left(3 \cdot s\right) + 0} \]

5. Simplified 1.3

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

   [Start] 1.3	\[ \log \left(\frac{1}{1 - \frac{u + -0.25}{0.75}}\right) \cdot \left(3 \cdot s\right) + 0 \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 1.3	\[ \color{blue}{\log \left(\frac{1}{1 - \frac{u + -0.25}{0.75}}\right) \cdot \left(3 \cdot s\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.3	\[ \color{blue}{3 \cdot \left(\log \left(\frac{1}{1 - \frac{u + -0.25}{0.75}}\right) \cdot s\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 1.3	\[ 3 \cdot \color{blue}{\left(s \cdot \log \left(\frac{1}{1 - \frac{u + -0.25}{0.75}}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.3	\[ \color{blue}{s \cdot \left(3 \cdot \log \left(\frac{1}{1 - \frac{u + -0.25}{0.75}}\right)\right)} \]

6. Final simplification 1.3

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

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	1.5
Cost	3552

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

Alternative 4
Error	1.5
Cost	3552

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

Alternative 5
Error	22.4
Cost	160

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

Alternative 6
Error	22.4
Cost	160

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

Reproduce

herbie shell --seed 2023090 
(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))))))