Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.3% → 98.2%

Time: 5.7s

Alternatives: 13

Speedup: 4.5×

Specification

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

Math

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

FPCore

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

C

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

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

MATLAB

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

MATLAB

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(u2 + u2\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt
     (* u1 (+ 1.0 (* u1 (+ 0.5 (* (fma u1 0.25 0.3333333333333333) u1))))))
    (sin (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (* (+ u2 u2) PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03999999910593033f) {
		tmp = sqrtf((u1 * (1.0f + (u1 * (0.5f + (fmaf(u1, 0.25f, 0.3333333333333333f) * u1)))))) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf(((u2 + u2) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03999999910593033))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(fma(u1, Float32(0.25), Float32(0.3333333333333333)) * u1)))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(u2 + u2) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.7%

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

   6. Applied rewrites 93.7%

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0399999991 < u1

   1. Initial program 57.3%

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

   3. Applied rewrites 57.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\left(u2 + u2\right) \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.017000000923871994:\\ \;\;\;\;\sqrt{-t\_0} \cdot \sin \left(\left(u2 + u2\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.017000000923871994)
     (* (sqrt (- t_0)) (sin (* (+ u2 u2) PI)))
     (*
      (sqrt (* u1 (+ 1.0 (* u1 (+ 0.5 (* 0.3333333333333333 u1))))))
      (sin (* (* 2.0 PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.017000000923871994f) {
		tmp = sqrtf(-t_0) * sinf(((u2 + u2) * ((float) M_PI)));
	} else {
		tmp = sqrtf((u1 * (1.0f + (u1 * (0.5f + (0.3333333333333333f * u1)))))) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.017000000923871994))
		tmp = Float32(sqrt(Float32(-t_0)) * sin(Float32(Float32(u2 + u2) * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(Float32(0.3333333333333333) * u1)))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(-0.017000000923871994))
		tmp = sqrt(-t_0) * sin(((u2 + u2) * single(pi)));
	else
		tmp = sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (single(0.3333333333333333) * u1)))))) * sin(((single(2.0) * single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0170000009

   1. Initial program 57.3%

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

   3. Applied rewrites 57.3%

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

   if -0.0170000009 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 92.0%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0035000001080334187:\\ \;\;\;\;\sqrt{-t\_0} \cdot \sin \left(\left(u2 + u2\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0035000001080334187)
     (* (sqrt (- t_0)) (sin (* (+ u2 u2) PI)))
     (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* (* 2.0 PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0035000001080334187f) {
		tmp = sqrtf(-t_0) * sinf(((u2 + u2) * ((float) M_PI)));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0035000001080334187))
		tmp = Float32(sqrt(Float32(-t_0)) * sin(Float32(Float32(u2 + u2) * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(-0.0035000001080334187))
		tmp = sqrt(-t_0) * sin(((u2 + u2) * single(pi)));
	else
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * sin(((single(2.0) * single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00350000011

   1. Initial program 57.3%

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

   3. Applied rewrites 57.3%

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

   if -0.00350000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03500000014901161)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (* 2.0 (* u2 PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03500000014901161f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * (2.0f * (u2 * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03500000014901161))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.03500000014901161))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * sin(((single(2.0) * single(pi)) * u2));
	else
		tmp = sqrt(-log((single(1.0) - u1))) * (single(2.0) * (u2 * single(pi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0350000001

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0350000001 < u1

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. Applied rewrites 50.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot u1\right)\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0004199999966658652)
   (*
    (sqrt
     (* u1 (+ 1.0 (* u1 (+ 0.5 (* (fma u1 0.25 0.3333333333333333) u1))))))
    (* 2.0 (* u2 PI)))
   (* (sqrt u1) (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0004199999966658652f) {
		tmp = sqrtf((u1 * (1.0f + (u1 * (0.5f + (fmaf(u1, 0.25f, 0.3333333333333333f) * u1)))))) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf(u1) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0004199999966658652))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(fma(u1, Float32(0.25), Float32(0.3333333333333333)) * u1)))))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 4.19999997e-4

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.7%

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

   6. Applied rewrites 93.7%

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right) \cdot u1\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   9. Applied rewrites 78.5%

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot u1\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]

   if 4.19999997e-4 < u2

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.9%

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.02199999988079071:\\ \;\;\;\;2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.02199999988079071)
   (*
    2.0
    (*
     u2
     (*
      PI
      (sqrt (- (* u1 (- (* u1 (- (* -0.3333333333333333 u1) 0.5)) 1.0)))))))
   (* (sqrt (- (log (- 1.0 u1)))) (* 2.0 (* u2 PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.02199999988079071f) {
		tmp = 2.0f * (u2 * (((float) M_PI) * sqrtf(-(u1 * ((u1 * ((-0.3333333333333333f * u1) - 0.5f)) - 1.0f)))));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * (2.0f * (u2 * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.02199999988079071))
		tmp = Float32(Float32(2.0) * Float32(u2 * Float32(Float32(pi) * sqrt(Float32(-Float32(u1 * Float32(Float32(u1 * Float32(Float32(Float32(-0.3333333333333333) * u1) - Float32(0.5))) - Float32(1.0))))))));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.02199999988079071))
		tmp = single(2.0) * (u2 * (single(pi) * sqrt(-(u1 * ((u1 * ((single(-0.3333333333333333) * u1) - single(0.5))) - single(1.0))))));
	else
		tmp = sqrt(-log((single(1.0) - u1))) * (single(2.0) * (u2 * single(pi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0219999999

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto 2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)\right) \]

   7. Applied rewrites 77.2%

      \[\leadsto 2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}\right)\right) \]

   if 0.0219999999 < u1

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. Applied rewrites 50.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(u2 \cdot \pi\right)\\ \mathbf{if}\;u1 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{u1 \cdot \left(u1 \cdot \left(0.5 + \frac{1}{u1}\right)\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* u2 PI))))
   (if (<= u1 0.0035000001080334187)
     (* (sqrt (* u1 (* u1 (+ 0.5 (/ 1.0 u1))))) t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (u2 * ((float) M_PI));
	float tmp;
	if (u1 <= 0.0035000001080334187f) {
		tmp = sqrtf((u1 * (u1 * (0.5f + (1.0f / u1))))) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(u2 * Float32(pi)))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0035000001080334187))
		tmp = Float32(sqrt(Float32(u1 * Float32(u1 * Float32(Float32(0.5) + Float32(Float32(1.0) / u1))))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = single(2.0) * (u2 * single(pi));
	tmp = single(0.0);
	if (u1 <= single(0.0035000001080334187))
		tmp = sqrt((u1 * (u1 * (single(0.5) + (single(1.0) / u1))))) * t_0;
	else
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00350000011

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around inf

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

   7. Applied rewrites 88.4%

      \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \color{blue}{\left(0.5 + \frac{1}{u1}\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + \frac{1}{u1}\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   10. Applied rewrites 74.8%

       \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(0.5 + \frac{1}{u1}\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]

   if 0.00350000011 < u1

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. Applied rewrites 50.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(u2 \cdot \pi\right)\\ \mathbf{if}\;u1 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* u2 PI))))
   (if (<= u1 0.0035000001080334187)
     (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (u2 * ((float) M_PI));
	float tmp;
	if (u1 <= 0.0035000001080334187f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(u2 * Float32(pi)))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0035000001080334187))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = single(2.0) * (u2 * single(pi));
	tmp = single(0.0);
	if (u1 <= single(0.0035000001080334187))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * t_0;
	else
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00350000011

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. Applied rewrites 74.8%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]

   if 0.00350000011 < u1

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. Applied rewrites 50.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0035000001080334187)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (* 2.0 (* u2 PI)))
   (* 2.0 (* u2 (* PI (sqrt (- (log (- 1.0 u1)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0035000001080334187f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = 2.0f * (u2 * (((float) M_PI) * sqrtf(-logf((1.0f - u1)))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0035000001080334187))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(Float32(2.0) * Float32(u2 * Float32(Float32(pi) * sqrt(Float32(-log(Float32(Float32(1.0) - u1)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.0035000001080334187))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * (single(2.0) * (u2 * single(pi)));
	else
		tmp = single(2.0) * (u2 * (single(pi) * sqrt(-log((single(1.0) - u1)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00350000011

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. Applied rewrites 74.8%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]

   if 0.00350000011 < u1

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot u1\right)\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* u1 (+ 1.0 (* u1 (+ 0.5 (* (fma u1 0.25 0.3333333333333333) u1))))))
  (* 2.0 (* u2 PI))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * (0.5f + (fmaf(u1, 0.25f, 0.3333333333333333f) * u1)))))) * (2.0f * (u2 * ((float) M_PI)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(fma(u1, Float32(0.25), Float32(0.3333333333333333)) * u1)))))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Applied rewrites 93.7%

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

6. Applied rewrites 93.7%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right) \cdot u1\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

9. Applied rewrites 78.5%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right) \cdot u1\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
10. Add Preprocessing

Alternative 11: 74.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (* 2.0 (* u2 PI))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (0.5f * u1)))) * (2.0f * (u2 * ((float) M_PI)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * (single(2.0) * (u2 * single(pi)));
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u1 around 0

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

4. Applied rewrites 88.4%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

7. Applied rewrites 74.8%

   \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
8. Add Preprocessing

Alternative 12: 66.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 2.0 (* u2 (* PI (sqrt u1)))))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(u2 * Float32(Float32(pi) * sqrt(u1))))
end

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

   \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{u1}}\right)\right) \]

7. Applied rewrites 66.8%

   \[\leadsto 2 \cdot \left(u2 \cdot \left(\pi \cdot \color{blue}{\sqrt{u1}}\right)\right) \]
8. Add Preprocessing

Alternative 13: 14.1% accurate, 53.8× speedup

Math

\[\begin{array}{l} \\ \pi \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 PI)

C

float code(float cosTheta_i, float u1, float u2) {
	return (float) M_PI;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(pi)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(pi);
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)\right)} \]

4. Applied rewrites 50.5%

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

6. Applied rewrites 14.1%

   \[\leadsto \color{blue}{\pi} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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))))