Average Error: 0.0 → 0.0
Time: 12.1s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\left(1 - ux\right) + ux \cdot maxCos \]
↓
\[\frac{ux}{ux} + \left(ux \cdot maxCos - ux\right) \]
\left(1 - ux\right) + ux \cdot maxCos
↓
\frac{ux}{ux} + \left(ux \cdot maxCos - ux\right)

(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))
↓
(FPCore (ux uy maxCos) :precision binary32 (+ (/ ux ux) (- (* ux maxCos) ux)))
float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}
↓
float code(float ux, float uy, float maxCos) {
	return (ux / ux) + ((ux * maxCos) - ux);
}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \]

  3. Applied add-exp-log_binary320.1

    \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}} \]

  4. Simplified0.0

    \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(maxCos \cdot ux - ux\right)}} \]

  5. Taylor expanded in maxCos around inf 3.8

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

  6. Simplified0.0

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

  7. Final simplification0.0

    \[\leadsto \frac{ux}{ux} + \left(ux \cdot maxCos - ux\right) \]

Reproduce

herbie shell --seed 2021275 
(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)))