Beckmann Sample, near normal, slope_y

Average Error: 13.4 → 0.6

Time: 19.7s

Precision: binary32

Cost: 20004

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9629999995231628:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(-1 \cdot \left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + {u1}^{2} \cdot -0.5\right)\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9629999995231628)
   (*
    (sqrt (- (log (- 1.0 u1))))
    (sin (* PI (* -1.0 (* (/ u2 (* 2.0 PI)) (* PI -4.0))))))
   (*
    (sqrt
     (-
      u1
      (+
       (* -0.25 (pow u1 4.0))
       (+ (* -0.3333333333333333 (pow u1 3.0)) (* (pow u1 2.0) -0.5)))))
    (sin (* 2.0 (* u2 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

↓

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9629999995231628f) {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf((((float) M_PI) * (-1.0f * ((u2 / (2.0f * ((float) M_PI))) * (((float) M_PI) * -4.0f)))));
	} else {
		tmp = sqrtf((u1 - ((-0.25f * powf(u1, 4.0f)) + ((-0.3333333333333333f * powf(u1, 3.0f)) + (powf(u1, 2.0f) * -0.5f))))) * sinf((2.0f * (u2 * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

↓

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9629999995231628))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(pi) * Float32(Float32(-1.0) * Float32(Float32(u2 / Float32(Float32(2.0) * Float32(pi))) * Float32(Float32(pi) * Float32(-4.0)))))));
	else
		tmp = Float32(sqrt(Float32(u1 - Float32(Float32(Float32(-0.25) * (u1 ^ Float32(4.0))) + Float32(Float32(Float32(-0.3333333333333333) * (u1 ^ Float32(3.0))) + Float32((u1 ^ Float32(2.0)) * Float32(-0.5)))))) * sin(Float32(Float32(2.0) * Float32(u2 * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

↓

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9629999995231628))
		tmp = sqrt(-log((single(1.0) - u1))) * sin((single(pi) * (single(-1.0) * ((u2 / (single(2.0) * single(pi))) * (single(pi) * single(-4.0))))));
	else
		tmp = sqrt((u1 - ((single(-0.25) * (u1 ^ single(4.0))) + ((single(-0.3333333333333333) * (u1 ^ single(3.0))) + ((u1 ^ single(2.0)) * single(-0.5)))))) * sin((single(2.0) * (u2 * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.963

   1. Initial program 0.8

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Applied egg-rr 0.9

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(0 \cdot \left(\left(2 \cdot \pi\right) \cdot \frac{u2}{2 \cdot \pi}\right) - \left(\left(2 \cdot \pi\right) \cdot \frac{u2}{2 \cdot \pi}\right) \cdot \left(\pi \cdot -2\right)\right)} \]

   3. Simplified 0.9

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\left(\left(2 \cdot \pi\right) \cdot \frac{u2}{2 \cdot \pi}\right) \cdot \left(-\pi \cdot -2\right)\right)} \]
      Proof

      [Start] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(0 \cdot \left(\left(2 \cdot \pi\right) \cdot \frac{u2}{2 \cdot \pi}\right) - \left(\left(2 \cdot \pi\right) \cdot \frac{u2}{2 \cdot \pi}\right) \cdot \left(\pi \cdot -2\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-102 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\left(\left(2 \cdot \pi\right) \cdot \frac{u2}{2 \cdot \pi}\right) \cdot \left(0 - \pi \cdot -2\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-5 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(\left(2 \cdot \pi\right) \cdot \frac{u2}{2 \cdot \pi}\right) \cdot \color{blue}{\left(-\pi \cdot -2\right)}\right) \]

   4. Applied egg-rr 0.9

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) + \left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) \cdot \pi\right)} \]

   5. Simplified 0.9

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \left(-1 \cdot \left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -4\right)\right)\right)\right)} \]
      Proof

      [Start] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) + \left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) \cdot \pi\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) \cdot \pi + \pi \cdot \left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \left(\left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) + \left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(\color{blue}{\left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) \cdot -1} + \left(-\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-94 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(\left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) \cdot -1 + \color{blue}{\left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) \cdot -1}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(\left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right) \cdot -1 + \color{blue}{-1 \cdot \left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right)}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \color{blue}{\left(-1 \cdot \left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right) + \frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(-1 \cdot \left(\color{blue}{\left(\pi \cdot -2\right) \cdot \frac{u2}{2 \cdot \pi}} + \frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2\right)\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(-1 \cdot \color{blue}{\left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -2 + \pi \cdot -2\right)\right)}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.9	\[ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(-1 \cdot \left(\frac{u2}{2 \cdot \pi} \cdot \left(\color{blue}{-2 \cdot \pi} + \pi \cdot -2\right)\right)\right)\right) \]

   if 0.963 < (-.f32 1 u1)

   1. Initial program 15.7

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0 0.5

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Simplified 0.5

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-u1\right) + \left(-0.5 \cdot {u1}^{2} + \left(-0.25 \cdot {u1}^{4} + -0.3333333333333333 \cdot {u1}^{3}\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      Proof

      [Start] 0.5	\[ \sqrt{-\left(-1 \cdot u1 + \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \sqrt{-\left(\color{blue}{u1 \cdot -1} + \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.5	\[ \sqrt{-\left(\color{blue}{\left(-u1\right)} + \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.5	\[ \sqrt{-\left(\left(-u1\right) + \left(-0.25 \cdot {u1}^{4} + \color{blue}{\left(-0.5 \cdot {u1}^{2} + -0.3333333333333333 \cdot {u1}^{3}\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.5	\[ \sqrt{-\left(\left(-u1\right) + \color{blue}{\left(-0.5 \cdot {u1}^{2} + \left(-0.25 \cdot {u1}^{4} + -0.3333333333333333 \cdot {u1}^{3}\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u2 around inf 0.5

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)}} \]

   5. Simplified 0.5

      \[\leadsto \color{blue}{\sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + {u1}^{2} \cdot -0.5\right)\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
      Proof

      [Start] 0.5	\[ \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \color{blue}{\sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.5	\[ \sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + \color{blue}{{u1}^{2} \cdot -0.5}\right)\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9629999995231628:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(-1 \cdot \left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + {u1}^{2} \cdot -0.5\right)\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	19812

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9879999756813049:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot \left(-1 \cdot \left(\frac{u2}{2 \cdot \pi} \cdot \left(\pi \cdot -4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 2
Error	0.7
Cost	16644

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9879999756813049:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{u1}^{2} \cdot 0.5 + \left(u1 + {u1}^{3} \cdot 0.3333333333333333\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	16644

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9879999756813049:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	13284

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9968000054359436:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - -0.5 \cdot {u1}^{2}} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 5
Error	2.8
Cost	13220

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 6
Error	7.6
Cost	9792

\[\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \]

Alternative 7
Error	10.8
Cost	6592

\[2 \cdot \left(u2 \cdot \left(\sqrt{u1} \cdot \pi\right)\right) \]

Reproduce

herbie shell --seed 2023090 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))