UniformSampleCone, z

Average Error: 0.0 → 0.0

Time: 1.1min

Precision: binary64

Cost: 448

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

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

↓

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

FPCore

(FPCore (ux uy maxCos) :precision binary64 (+ (- 1.0 ux) (* ux maxCos)))

↓

(FPCore (ux uy maxCos) :precision binary64 (- (- (* ux maxCos) ux) -1.0))

C

double code(double ux, double uy, double maxCos) {
	return (1.0 - ux) + (ux * maxCos);
}

↓

double code(double ux, double uy, double maxCos) {
	return ((ux * maxCos) - ux) - -1.0;
}

Fortran

real(8) function code(ux, uy, maxcos)
    real(8), intent (in) :: ux
    real(8), intent (in) :: uy
    real(8), intent (in) :: maxcos
    code = (1.0d0 - ux) + (ux * maxcos)
end function

↓

real(8) function code(ux, uy, maxcos)
    real(8), intent (in) :: ux
    real(8), intent (in) :: uy
    real(8), intent (in) :: maxcos
    code = ((ux * maxcos) - ux) - (-1.0d0)
end function

Java

public static double code(double ux, double uy, double maxCos) {
	return (1.0 - ux) + (ux * maxCos);
}

↓

public static double code(double ux, double uy, double maxCos) {
	return ((ux * maxCos) - ux) - -1.0;
}

Python

def code(ux, uy, maxCos):
	return (1.0 - ux) + (ux * maxCos)

↓

def code(ux, uy, maxCos):
	return ((ux * maxCos) - ux) - -1.0

Julia

function code(ux, uy, maxCos)
	return Float64(Float64(1.0 - ux) + Float64(ux * maxCos))
end

↓

function code(ux, uy, maxCos)
	return Float64(Float64(Float64(ux * maxCos) - ux) - -1.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (1.0 - ux) + (ux * maxCos);
end

↓

function tmp = code(ux, uy, maxCos)
	tmp = ((ux * maxCos) - ux) - -1.0;
end

Wolfram

code[ux_, uy_, maxCos_] := N[(N[(1.0 - ux), $MachinePrecision] + N[(ux * maxCos), $MachinePrecision]), $MachinePrecision]

↓

code[ux_, uy_, maxCos_] := N[(N[(N[(ux * maxCos), $MachinePrecision] - ux), $MachinePrecision] - -1.0), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

3. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	448

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

Alternative 2
Error	0.8
Cost	192

\[1 - ux \]

Alternative 3
Error	37.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, z"
  :precision binary64
  :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)))