UniformSampleCone 2

Average Error: 0.3 → 0.3

Time: 23.2s

Precision: binary32

Cost: 52448

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

↓

\[\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(ux + -1\right) \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos \left(\log \left(\sqrt{{\left(e^{2 \cdot \pi}\right)}^{uy}}\right) + \log \left(\sqrt{e^{2 \cdot \left(\pi \cdot uy\right)}}\right)\right), xi, yi \cdot \sin \left(\left(2 \cdot \pi\right) \cdot uy\right)\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  (+
   (*
    (*
     (cos (* (* uy 2.0) PI))
     (sqrt
      (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux)))))
    xi)
   (*
    (*
     (sin (* (* uy 2.0) PI))
     (sqrt
      (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux)))))
    yi))
  (* (* (* (- 1.0 ux) maxCos) ux) zi)))

↓

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (sqrt (fma ux (* (+ ux -1.0) (* maxCos (* (- 1.0 ux) (* ux maxCos)))) 1.0))
  (fma
   (cos
    (+
     (log (sqrt (pow (exp (* 2.0 PI)) uy)))
     (log (sqrt (exp (* 2.0 (* PI uy)))))))
   xi
   (* yi (sin (* (* 2.0 PI) uy))))
  (* ux (* (- 1.0 ux) (* maxCos zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - ((((1.0f - ux) * maxCos) * ux) * (((1.0f - ux) * maxCos) * ux))))) * xi) + ((sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - ((((1.0f - ux) * maxCos) * ux) * (((1.0f - ux) * maxCos) * ux))))) * yi)) + ((((1.0f - ux) * maxCos) * ux) * zi);
}

↓

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(sqrtf(fmaf(ux, ((ux + -1.0f) * (maxCos * ((1.0f - ux) * (ux * maxCos)))), 1.0f)), fmaf(cosf((logf(sqrtf(powf(expf((2.0f * ((float) M_PI))), uy))) + logf(sqrtf(expf((2.0f * (((float) M_PI) * uy))))))), xi, (yi * sinf(((2.0f * ((float) M_PI)) * uy)))), (ux * ((1.0f - ux) * (maxCos * zi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux))))) * xi) + Float32(Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux))))) * yi)) + Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi))
end

↓

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(sqrt(fma(ux, Float32(Float32(ux + Float32(-1.0)) * Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos)))), Float32(1.0))), fma(cos(Float32(log(sqrt((exp(Float32(Float32(2.0) * Float32(pi))) ^ uy))) + log(sqrt(exp(Float32(Float32(2.0) * Float32(Float32(pi) * uy))))))), xi, Float32(yi * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * uy)))), Float32(ux * Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * zi))))
end

Derivation

1. Initial program 0.3

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(ux + -1\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), xi, \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right)} \]
   Proof
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (*.f32 (*.f32 ux maxCos) (+.f32 ux -1)))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 maxCos ux)) (+.f32 ux -1)))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (*.f32 (*.f32 maxCos ux) (Rewrite<= +-commutative_binary32 (+.f32 -1 ux))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (*.f32 (*.f32 maxCos ux) (+.f32 (Rewrite<= metadata-eval (-.f32 0 1)) ux)))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (*.f32 (*.f32 maxCos ux) (Rewrite<= associate--r-_binary32 (-.f32 0 (-.f32 1 ux)))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (*.f32 (*.f32 maxCos ux) (Rewrite<= neg-sub0_binary32 (neg.f32 (-.f32 1 ux)))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 (*.f32 maxCos ux) (-.f32 1 ux)))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (neg.f32 (Rewrite<= *-commutative_binary32 (*.f32 (-.f32 1 ux) (*.f32 maxCos ux)))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos (neg.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (Rewrite=> distribute-rgt-neg-out_binary32 (neg.f32 (*.f32 maxCos (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (neg.f32 (*.f32 maxCos (Rewrite=> *-commutative_binary32 (*.f32 ux (*.f32 (-.f32 1 ux) maxCos)))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (*.f32 (-.f32 1 ux) (neg.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 maxCos ux) (*.f32 (-.f32 1 ux) maxCos))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 (-.f32 1 ux) (*.f32 (*.f32 maxCos ux) (*.f32 (-.f32 1 ux) maxCos))))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (neg.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (-.f32 1 ux) (*.f32 maxCos ux)) (*.f32 (-.f32 1 ux) maxCos)))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (neg.f32 (*.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)) (*.f32 (-.f32 1 ux) maxCos))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (fma.f32 ux (Rewrite<= distribute-lft-neg-out_binary32 (*.f32 (neg.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)) (*.f32 (-.f32 1 ux) maxCos))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 ux (*.f32 (neg.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)) (*.f32 (-.f32 1 ux) maxCos))) 1))) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 (*.f32 (neg.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)) (*.f32 (-.f32 1 ux) maxCos)) ux)) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (+.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (neg.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux))) 1)) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 (neg.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux))))) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (Rewrite<= cancel-sign-sub-inv_binary32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux))))) (fma.f32 (cos.f32 (*.f32 uy (*.f32 2 (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (fma.f32 (cos.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 uy 2) (PI.f32)))) xi (*.f32 (sin.f32 (*.f32 uy (*.f32 2 (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (fma.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) xi (*.f32 (sin.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 uy 2) (PI.f32)))) yi)) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 0 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) xi) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi))) (*.f32 ux (*.f32 (-.f32 1 ux) (*.f32 maxCos zi)))): 3 points increase in error, 0 points decrease in error
   (fma.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (+.f32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) xi) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi)) (*.f32 ux (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) zi)))): 2 points increase in error, 2 points decrease in error
   (fma.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (+.f32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) xi) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi)) (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 ux (*.f32 (-.f32 1 ux) maxCos)) zi))): 4 points increase in error, 3 points decrease in error
   (fma.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (+.f32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) xi) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi)) (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)) zi)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (+.f32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) xi) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi))) (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) zi))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= distribute-lft-out_binary32 (+.f32 (*.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) xi)) (*.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi)))) (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) zi)): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))) xi)) (*.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi))) (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) zi)): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))))) xi) (*.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) yi))) (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) zi)): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux))))) xi) (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))) (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32)))) yi))) (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) zi)): 0 points increase in error, 0 points decrease in error
   (+.f32 (+.f32 (*.f32 (*.f32 (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux))))) xi) (*.f32 (Rewrite<= *-commutative_binary32 (*.f32 (sin.f32 (*.f32 (*.f32 uy 2) (PI.f32))) (sqrt.f32 (-.f32 1 (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux)))))) yi)) (*.f32 (*.f32 (*.f32 (-.f32 1 ux) maxCos) ux) zi)): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.3

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(ux + -1\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos \color{blue}{\left(\log \left(\sqrt{{\left({\left(e^{2}\right)}^{\pi}\right)}^{uy}}\right) + \log \left(\sqrt{{\left({\left(e^{2}\right)}^{\pi}\right)}^{uy}}\right)\right)}, xi, \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right) \]

4. Applied egg-rr 0.3

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(ux + -1\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos \left(\log \left(\sqrt{{\left({\left(e^{2}\right)}^{\pi}\right)}^{uy}}\right) + \log \left(\sqrt{{\color{blue}{\left(e^{2 \cdot \pi}\right)}}^{uy}}\right)\right), xi, \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right) \]

5. Taylor expanded in uy around inf 0.3

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(ux + -1\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos \left(\log \left(\sqrt{{\left({\left(e^{2}\right)}^{\pi}\right)}^{uy}}\right) + \color{blue}{\log \left(\sqrt{e^{2 \cdot \left(uy \cdot \pi\right)}}\right)}\right), xi, \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right) \]

6. Applied egg-rr 0.3

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(ux + -1\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos \left(\log \left(\sqrt{{\color{blue}{\left(e^{2 \cdot \pi}\right)}}^{uy}}\right) + \log \left(\sqrt{e^{2 \cdot \left(uy \cdot \pi\right)}}\right)\right), xi, \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right) \]

7. Final simplification 0.3

   \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(ux + -1\right) \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos \left(\log \left(\sqrt{{\left(e^{2 \cdot \pi}\right)}^{uy}}\right) + \log \left(\sqrt{e^{2 \cdot \left(\pi \cdot uy\right)}}\right)\right), xi, yi \cdot \sin \left(\left(2 \cdot \pi\right) \cdot uy\right)\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	26720

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot uy\\ \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(ux + -1\right) \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right), 1\right)}, \mathsf{fma}\left(\cos t_0, xi, yi \cdot \sin t_0\right), ux \cdot \left(\left(1 - ux\right) \cdot \left(maxCos \cdot zi\right)\right)\right) \end{array} \]

Alternative 2
Error	0.4
Cost	17696

\[\left(xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) + \sin \left(\left(2 \cdot \pi\right) \cdot uy\right) \cdot \left(yi + \left(yi \cdot -0.5\right) \cdot \left(maxCos \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)\right)\right)\right) + zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \]

Alternative 3
Error	0.4
Cost	17312

\[zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + \left(xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) + yi \cdot \sin \left(\left(2 \cdot \pi\right) \cdot uy\right)\right) \]

Alternative 4
Error	0.4
Cost	17056

\[\left(yi \cdot \sin \left(\left(2 \cdot \pi\right) \cdot uy\right) + xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - maxCos \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)}\right)\right) + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \]

Alternative 5
Error	3.2
Cost	14112

\[\begin{array}{l} t_0 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ zi \cdot t_0 + \left(xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + t_0 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) + \left(\left(2 \cdot \pi\right) \cdot uy\right) \cdot yi\right) \end{array} \]

Alternative 6
Error	3.2
Cost	14112

\[\begin{array}{l} t_0 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ zi \cdot t_0 + \left(xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + t_0 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \end{array} \]

Alternative 7
Error	3.2
Cost	13856

\[zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) + \left(\pi \cdot \left(2 \cdot \left(uy \cdot yi\right)\right) + xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - maxCos \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)}\right)\right) \]

Alternative 8
Error	13.0
Cost	10688

\[xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) + ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) \]

Alternative 9
Error	13.0
Cost	10688

\[\begin{array}{l} t_0 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + t_0 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) + zi \cdot t_0 \end{array} \]

Alternative 10
Error	13.0
Cost	10560

\[zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) + xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + \left(ux \cdot maxCos\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) \]

Alternative 11
Error	13.0
Cost	10432

\[ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - maxCos \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)}\right) \]

Alternative 12
Error	13.0
Cost	10432

\[zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) + xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - maxCos \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)}\right) \]

Alternative 13
Error	13.7
Cost	10304

\[xi \cdot \left(\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - maxCos \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)}\right) + \left(ux \cdot maxCos\right) \cdot zi \]

Reproduce

herbie shell --seed 2022308 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))