UniformSampleCone, z

Average Accuracy: 99.9% → 99.9%

Time: 7.2s

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

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}\right)}^{3}} \]
   Proof

   [Start] 99.9	\[ \left(1 - ux\right) + ux \cdot maxCos \]
   add-cube-cbrt [=>] 98.2	\[ \color{blue}{\left(\sqrt[3]{\left(1 - ux\right) + ux \cdot maxCos} \cdot \sqrt[3]{\left(1 - ux\right) + ux \cdot maxCos}\right) \cdot \sqrt[3]{\left(1 - ux\right) + ux \cdot maxCos}} \]
   pow3 [=>] 98.2	\[ \color{blue}{{\left(\sqrt[3]{\left(1 - ux\right) + ux \cdot maxCos}\right)}^{3}} \]
   +-commutative [=>] 98.2	\[ {\left(\sqrt[3]{\color{blue}{ux \cdot maxCos + \left(1 - ux\right)}}\right)}^{3} \]
   fma-def [=>] 98.2	\[ {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}}\right)}^{3} \]

3. Applied egg-rr 99.9%

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

   [Start] 98.2	\[ {\left(\sqrt[3]{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}\right)}^{3} \]
   rem-cube-cbrt [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \]
   fma-udef [=>] 99.9	\[ \color{blue}{ux \cdot maxCos + \left(1 - ux\right)} \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(1 - ux\right) + ux \cdot maxCos} \]
   associate-+l- [=>] 99.9	\[ \color{blue}{1 - \left(ux - ux \cdot maxCos\right)} \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	224

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

Alternative 2
Accuracy	99.9%
Cost	224

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

Alternative 3
Accuracy	98.1%
Cost	96

\[1 - ux \]

Alternative 4
Accuracy	72.0%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023138 
(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)))