UniformSampleCone, z

Average Error: 0.0 → 0.0

Time: 2.8s

Precision: binary32

Cost: 224

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

Math

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

↓

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

FPCore

(FPCore (ux uy maxCos) :precision binary32 (+ (- 1.0 ux) (* ux maxCos)))

↓

(FPCore (ux uy maxCos) :precision binary32 (- 1.0 (* ux (- 1.0 maxCos))))

C

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 - (ux * (1.0f - maxCos));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (1.0e0 - ux) + (ux * maxcos)
end function

↓

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 - (ux * (1.0e0 - maxcos))
end function

Julia

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) - Float32(ux * Float32(Float32(1.0) - maxCos)))
end

MATLAB

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

↓

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

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. Simplified 0.0

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

   [Start] 0.0	\[ 1 + \left(maxCos - 1\right) \cdot ux \]
   rational.json-simplify-17 [=>] 0.0	\[ \color{blue}{\left(maxCos - 1\right) \cdot ux - -1} \]
   rational.json-simplify-2 [=>] 0.0	\[ \color{blue}{ux \cdot \left(maxCos - 1\right)} - -1 \]
   metadata-eval [<=] 0.0	\[ ux \cdot \left(maxCos - \color{blue}{\left(1 - 0\right)}\right) - -1 \]
   rational.json-simplify-45 [<=] 0.0	\[ ux \cdot \color{blue}{\left(0 - \left(1 - maxCos\right)\right)} - -1 \]
   rational.json-simplify-27 [<=] 0.0	\[ ux \cdot \left(0 - \color{blue}{\left(\left(0 - maxCos\right) + \left(1 - 0\right)\right)}\right) - -1 \]
   rational.json-simplify-12 [<=] 0.0	\[ ux \cdot \left(0 - \left(\color{blue}{\left(-maxCos\right)} + \left(1 - 0\right)\right)\right) - -1 \]
   rational.json-simplify-9 [<=] 0.0	\[ ux \cdot \left(0 - \left(\color{blue}{maxCos \cdot -1} + \left(1 - 0\right)\right)\right) - -1 \]
   rational.json-simplify-2 [<=] 0.0	\[ ux \cdot \left(0 - \left(\color{blue}{-1 \cdot maxCos} + \left(1 - 0\right)\right)\right) - -1 \]
   metadata-eval [=>] 0.0	\[ ux \cdot \left(0 - \left(-1 \cdot maxCos + \color{blue}{1}\right)\right) - -1 \]
   rational.json-simplify-1 [<=] 0.0	\[ ux \cdot \left(0 - \color{blue}{\left(1 + -1 \cdot maxCos\right)}\right) - -1 \]
   rational.json-simplify-12 [<=] 0.0	\[ ux \cdot \color{blue}{\left(-\left(1 + -1 \cdot maxCos\right)\right)} - -1 \]
   rational.json-simplify-9 [<=] 0.0	\[ ux \cdot \color{blue}{\left(\left(1 + -1 \cdot maxCos\right) \cdot -1\right)} - -1 \]
   rational.json-simplify-43 [<=] 0.0	\[ \color{blue}{-1 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right)} - -1 \]
   rational.json-simplify-2 [=>] 0.0	\[ \color{blue}{\left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) \cdot -1} - -1 \]
   rational.json-simplify-9 [=>] 0.0	\[ \color{blue}{\left(-ux \cdot \left(1 + -1 \cdot maxCos\right)\right)} - -1 \]
   rational.json-simplify-12 [=>] 0.0	\[ \color{blue}{\left(0 - ux \cdot \left(1 + -1 \cdot maxCos\right)\right)} - -1 \]
   rational.json-simplify-42 [=>] 0.0	\[ \color{blue}{\left(0 - -1\right) - ux \cdot \left(1 + -1 \cdot maxCos\right)} \]
   metadata-eval [=>] 0.0	\[ \color{blue}{1} - ux \cdot \left(1 + -1 \cdot maxCos\right) \]
   rational.json-simplify-1 [=>] 0.0	\[ 1 - ux \cdot \color{blue}{\left(-1 \cdot maxCos + 1\right)} \]
   rational.json-simplify-2 [=>] 0.0	\[ 1 - ux \cdot \left(\color{blue}{maxCos \cdot -1} + 1\right) \]
   rational.json-simplify-9 [=>] 0.0	\[ 1 - ux \cdot \left(\color{blue}{\left(-maxCos\right)} + 1\right) \]
   rational.json-simplify-12 [=>] 0.0	\[ 1 - ux \cdot \left(\color{blue}{\left(0 - maxCos\right)} + 1\right) \]
   metadata-eval [<=] 0.0	\[ 1 - ux \cdot \left(\left(0 - maxCos\right) + \color{blue}{\left(1 - 0\right)}\right) \]
   rational.json-simplify-27 [=>] 0.0	\[ 1 - ux \cdot \color{blue}{\left(1 - maxCos\right)} \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	224

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

Alternative 2
Error	0.6
Cost	96

\[1 - ux \]

Alternative 3
Error	9.1
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023074 
(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)))