Average Error: 0.0 → 0.1
Time: 3.1s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\left(1 - ux\right) + ux \cdot maxCos \]
↓
\[1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\left(maxCos + -1\right) \cdot ux\right)\right) \]
(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))
↓
(FPCore (ux uy maxCos)
 :precision binary32
 (+ 1.0 (log1p (expm1 (* (+ maxCos -1.0) ux)))))
float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}
↓
float code(float ux, float uy, float maxCos) {
	return 1.0f + log1pf(expm1f(((maxCos + -1.0f) * ux)));
}

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end
↓
function code(ux, uy, maxCos)
	return Float32(Float32(1.0) + log1p(expm1(Float32(Float32(maxCos + Float32(-1.0)) * ux))))
end

\left(1 - ux\right) + ux \cdot maxCos
↓
1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\left(maxCos + -1\right) \cdot ux\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(1 - ux\right) + ux \cdot maxCos \]

  2. Taylor expanded in ux around 0 0.0

    \[\leadsto \color{blue}{1 + \left(maxCos - 1\right) \cdot ux} \]

  3. Applied egg-rr0.1

    \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(maxCos + -1\right) \cdot ux\right)\right)} \]

  4. Final simplification0.1

    \[\leadsto 1 + \mathsf{log1p}\left(\mathsf{expm1}\left(\left(maxCos + -1\right) \cdot ux\right)\right) \]

Reproduce

herbie shell --seed 2022211 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, z"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))