Average Error: 1.4 → 0.5
Time: 13.7s
Precision: binary32
Cost: 3488
\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]
\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
↓
\[s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \]
(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 (log1p (/ (- 0.25 u) 0.75)))))
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 * log1pf(((0.25f - u) / 0.75f)));
}

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

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

Error

Try it out

Your Program's Arguments

Results

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Derivation

  1. Initial program 1.4

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

  2. Simplified0.5

    \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.25 + \left(-u\right)}{0.75}\right)\right)} \]
    Proof
    (*.f32 s (*.f32 -3 (log1p.f32 (/.f32 (+.f32 1/4 (neg.f32 u)) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 (Rewrite<= metadata-eval (*.f32 3 -1)) (log1p.f32 (/.f32 (+.f32 1/4 (neg.f32 u)) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 (*.f32 3 -1) (log1p.f32 (/.f32 (+.f32 (Rewrite<= metadata-eval (neg.f32 -1/4)) (neg.f32 u)) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 (*.f32 3 -1) (log1p.f32 (/.f32 (+.f32 (neg.f32 (Rewrite<= metadata-eval (neg.f32 1/4))) (neg.f32 u)) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 (*.f32 3 -1) (log1p.f32 (/.f32 (Rewrite<= distribute-neg-in_binary32 (neg.f32 (+.f32 (neg.f32 1/4) u))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 (*.f32 3 -1) (log1p.f32 (/.f32 (neg.f32 (Rewrite<= +-commutative_binary32 (+.f32 u (neg.f32 1/4)))) 3/4)))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 (*.f32 3 -1) (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 s (*.f32 (*.f32 3 -1) (log1p.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4)))))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 (*.f32 3 -1) (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4))))))): 93 points increase in error, 2 points decrease in error
    (*.f32 s (*.f32 (*.f32 3 -1) (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 s (Rewrite<= associate-*r*_binary32 (*.f32 3 (*.f32 -1 (log.f32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 3 (Rewrite<= neg-mul-1_binary32 (neg.f32 (log.f32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))))): 0 points increase in error, 0 points decrease in error
    (*.f32 s (*.f32 3 (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))))): 67 points increase in error, 37 points decrease in error
    (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 s 3) (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))))): 46 points increase in error, 51 points decrease in error
    (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 3 s)) (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.5

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

Alternatives

Alternative 1
Error1.0
Cost3488
\[\mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right) \cdot \left(s \cdot -3\right) \]
Alternative 2
Error0.7
Cost3488
\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right) \]
Alternative 3
Error0.7
Cost3488
\[s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right) \]
Alternative 4
Error22.4
Cost160
\[3 \cdot \left(s \cdot u\right) \]

Error

Reproduce

herbie shell --seed 2022296 
(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))))))