Average Error: 1.3 → 0.5
Time: 12.6s
Precision: binary32
Cost: 3488
\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]
\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
↓
\[\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot \left(s \cdot -3\right) \]
(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))
↓
(FPCore (s u) :precision binary32 (* (log1p (/ (- 0.25 u) 0.75)) (* s -3.0)))
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 log1pf(((0.25f - u) / 0.75f)) * (s * -3.0f);
}

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(log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75))) * Float32(s * Float32(-3.0)))
end

\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)
↓
\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot \left(s \cdot -3\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.5

    \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\frac{-\left(u + -0.25\right)}{0.75}\right)\right)} \]
    Proof
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (+.f32 u -1/4)) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (+.f32 u (Rewrite<= metadata-eval (neg.f32 1/4)))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 u 1/4))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (Rewrite<= --rgt-identity_binary32 (-.f32 (-.f32 u 1/4) 0))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (-.f32 (-.f32 u 1/4) (Rewrite<= metadata-eval (log.f32 1)))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (Rewrite<= sub0-neg_binary32 (-.f32 0 (-.f32 (-.f32 u 1/4) (log.f32 1)))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (-.f32 (Rewrite<= metadata-eval (log.f32 1)) (-.f32 (-.f32 u 1/4) (log.f32 1))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (log.f32 1) 3/4) (/.f32 (-.f32 (-.f32 u 1/4) (log.f32 1)) 3/4)))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 (/.f32 (Rewrite=> metadata-eval 0) 3/4) (/.f32 (-.f32 (-.f32 u 1/4) (log.f32 1)) 3/4))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 (Rewrite=> metadata-eval 0) (/.f32 (-.f32 (-.f32 u 1/4) (log.f32 1)) 3/4))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 0 (/.f32 (-.f32 (-.f32 u 1/4) (Rewrite=> metadata-eval 0)) 3/4))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 0 (/.f32 (Rewrite=> --rgt-identity_binary32 (-.f32 u 1/4)) 3/4))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (Rewrite<= neg-sub0_binary32 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4)))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4))))))): 97 points increase in error, 6 points decrease in error
    (*.f32 (*.f32 3 s) (neg.f32 (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))))): 0 points increase in error, 0 points decrease in error
    (*.f32 (*.f32 3 s) (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))))): 85 points increase in error, 43 points decrease in error

  3. Final simplification0.5

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

Alternatives

Alternative 1
Error1.2
Cost3488
\[-3 \cdot \left(s \cdot \log \left(1.3333333333333333 - u \cdot 1.3333333333333333\right)\right) \]
Alternative 2
Error0.7
Cost3488
\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot -1.3333333333333333\right)\right) \]
Alternative 3
Error0.7
Cost3488
\[\mathsf{log1p}\left(\left(u + -0.25\right) \cdot -1.3333333333333333\right) \cdot \left(s \cdot -3\right) \]
Alternative 4
Error0.6
Cost3488
\[s \cdot \left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot -3\right) \]
Alternative 5
Error26.0
Cost3424
\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(u \cdot -1.3333333333333333\right)\right) \]
Alternative 6
Error23.8
Cost3424
\[-3 \cdot \left(s \cdot \left(\log 1.3333333333333333 - u\right)\right) \]
Alternative 7
Error23.8
Cost3424
\[s \cdot \left(-3 \cdot \left(\log 1.3333333333333333 - u\right)\right) \]
Alternative 8
Error23.8
Cost3424
\[s \cdot \left(3 \cdot \left(u + \log 0.75\right)\right) \]
Alternative 9
Error28.6
Cost32
\[0 \]

Error

Reproduce

herbie shell --seed 2022326 
(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))))))