Beckmann Sample, near normal, slope_y

Average Error: 13.3 → 0.6

Time: 24.1s

Precision: binary32

Cost: 20068

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.949999988079071:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-u1\right) + \left(-0.5 \cdot {u1}^{2} + \left(-0.25 \cdot {u1}^{4} + -0.3333333333333333 \cdot {u1}^{3}\right)\right)\right)} \cdot t_0\\ \end{array} \]

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
 (let* ((t_0 (sin (* (* 2.0 PI) u2))))
   (if (<= (- 1.0 u1) 0.949999988079071)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (*
      (sqrt
       (-
        (+
         (- u1)
         (+
          (* -0.5 (pow u1 2.0))
          (+ (* -0.25 (pow u1 4.0)) (* -0.3333333333333333 (pow u1 3.0)))))))
      t_0))))

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 t_0 = sinf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if ((1.0f - u1) <= 0.949999988079071f) {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	} else {
		tmp = sqrtf(-(-u1 + ((-0.5f * powf(u1, 2.0f)) + ((-0.25f * powf(u1, 4.0f)) + (-0.3333333333333333f * powf(u1, 3.0f)))))) * t_0;
	}
	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)
	t_0 = sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.949999988079071))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-Float32(Float32(-u1) + Float32(Float32(Float32(-0.5) * (u1 ^ Float32(2.0))) + Float32(Float32(Float32(-0.25) * (u1 ^ Float32(4.0))) + Float32(Float32(-0.3333333333333333) * (u1 ^ Float32(3.0)))))))) * t_0);
	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)
	t_0 = sin(((single(2.0) * single(pi)) * u2));
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.949999988079071))
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	else
		tmp = sqrt(-(-u1 + ((single(-0.5) * (u1 ^ single(2.0))) + ((single(-0.25) * (u1 ^ single(4.0))) + (single(-0.3333333333333333) * (u1 ^ single(3.0))))))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.949999988

   1. Initial program 0.7

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

   if 0.949999988 < (-.f32 1 u1)

   1. Initial program 15.2

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

   2. Taylor expanded in u1 around 0 0.6

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

   3. Simplified 0.6

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

      [Start] 0.6	\[ \sqrt{-\left(-1 \cdot u1 + \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational.json-simplify-50 [=>] 0.6	\[ \sqrt{-\left(\color{blue}{u1 \cdot -1} + \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational.json-simplify-38 [=>] 0.6	\[ \sqrt{-\left(\color{blue}{\left(-u1\right)} + \left(-0.25 \cdot {u1}^{4} + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational.json-simplify-1 [=>] 0.6	\[ \sqrt{-\left(\left(-u1\right) + \left(-0.25 \cdot {u1}^{4} + \color{blue}{\left(-0.5 \cdot {u1}^{2} + -0.3333333333333333 \cdot {u1}^{3}\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
      rational.json-simplify-2 [=>] 0.6	\[ \sqrt{-\left(\left(-u1\right) + \color{blue}{\left(-0.5 \cdot {u1}^{2} + \left(-0.25 \cdot {u1}^{4} + -0.3333333333333333 \cdot {u1}^{3}\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.949999988079071:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-u1\right) + \left(-0.5 \cdot {u1}^{2} + \left(-0.25 \cdot {u1}^{4} + -0.3333333333333333 \cdot {u1}^{3}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	20004

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9599999785423279:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{u1}^{4} \cdot 0.25 + \left(0.3333333333333333 \cdot {u1}^{3} + \left(u1 + 0.5 \cdot {u1}^{2}\right)\right)} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	20004

\[\begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.949999988079071:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - \left(-0.25 \cdot {u1}^{4} + -0.5 \cdot \left({u1}^{2} + {u1}^{3} \cdot 0.6666666666666666\right)\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	16708

\[\begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9800000190734863:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-u1\right) + \left(-0.5 \cdot {u1}^{2} + -0.3333333333333333 \cdot {u1}^{3}\right)\right)} \cdot t_0\\ \end{array} \]

Alternative 4
Error	2.8
Cost	16452

\[\begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t_0 \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \sin \left(u2 + \left(2 \cdot \left(u2 \cdot \pi\right) - u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]

Alternative 5
Error	4.6
Cost	13348

\[\begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t_0 \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \sin \left(u2 + \left(2 \cdot \left(u2 \cdot \pi\right) - u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\frac{\frac{0.5}{u2}}{\pi}}\\ \end{array} \]

Alternative 6
Error	1.0
Cost	13348

\[\begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9959999918937683:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-u1\right) + -0.5 \cdot {u1}^{2}\right)} \cdot t_0\\ \end{array} \]

Alternative 7
Error	4.6
Cost	13252

\[\begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t_0 \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\frac{0.5}{u2 \cdot \pi}}\\ \end{array} \]

Alternative 8
Error	4.6
Cost	13252

\[\begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t_0 \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\frac{\frac{0.5}{u2}}{\pi}}\\ \end{array} \]

Alternative 9
Error	7.7
Cost	9792

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

Alternative 10
Error	10.8
Cost	6592

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

Alternative 11
Error	10.8
Cost	6592

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

Reproduce

herbie shell --seed 2023073 
(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))))