Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.8% → 98.7%

Time: 5.2s

Alternatives: 13

Speedup: 21.0×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\pi \cdot 0.5 - -2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03500000014901161)
   (*
    (sqrt
     (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1))
    (cos (* (+ PI PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (- (* PI 0.5) (* -2.0 (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03500000014901161f) {
		tmp = sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 1.0f) * u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf(((((float) M_PI) * 0.5f) - (-2.0f * (((float) M_PI) * u2))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03500000014901161))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(pi) * Float32(0.5)) - Float32(Float32(-2.0) * Float32(Float32(pi) * u2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.03500000014901161))
		tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * cos(((single(pi) + single(pi)) * u2));
	else
		tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(pi) * single(0.5)) - (single(-2.0) * (single(pi) * u2))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0350000001

   1. Initial program 54.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.9

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

   5. Applied rewrites 98.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 98.9

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

   7. Applied rewrites 98.9%

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

   if 0.0350000001 < u1

   1. Initial program 97.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f32 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lift-PI.f32 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 97.2

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

   7. Applied rewrites 97.2%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (+ PI PI) u2))))
   (if (<= u1 0.03500000014901161)
     (*
      (sqrt
       (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 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 = cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.03500000014901161f) {
		tmp = sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 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 = cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03500000014901161))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + 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 = cos(((single(pi) + single(pi)) * u2));
	tmp = single(0.0);
	if (u1 <= single(0.03500000014901161))
		tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + 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 < 0.0350000001

   1. Initial program 54.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.9

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

   5. Applied rewrites 98.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 98.9

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

   7. Applied rewrites 98.9%

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

   if 0.0350000001 < u1

   1. Initial program 97.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 97.0

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

   4. Applied rewrites 97.0%

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

Alternative 3: 93.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * cos(((single(pi) + single(pi)) * u2));
end

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 93.8

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

5. Applied rewrites 93.8%

   \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 93.8

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

7. Applied rewrites 93.8%

   \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \cos \left(\color{blue}{\left(\pi + \pi\right)} \cdot u2\right) \]
8. Add Preprocessing

Alternative 4: 92.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((((single(0.3333333333333333) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * cos(((single(pi) + single(pi)) * u2));
end

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 91.3

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

5. Applied rewrites 91.3%

   \[\leadsto \sqrt{\color{blue}{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 91.3

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

7. Applied rewrites 91.3%

   \[\leadsto \sqrt{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \cos \left(\color{blue}{\left(\pi + \pi\right)} \cdot u2\right) \]
8. Add Preprocessing

Alternative 5: 93.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.04500000178813934:\\ \;\;\;\;\sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(0.5 \cdot u1 + 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.04500000178813934)
   (*
    (sqrt
     (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1))
    (+
     (*
      (+
       (* (* 0.6666666666666666 (* u2 u2)) (* (* PI PI) (* PI PI)))
       (* (* PI PI) -2.0))
      (* u2 u2))
     1.0))
   (* (sqrt (* (+ (* 0.5 u1) 1.0) u1)) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.04500000178813934f) {
		tmp = sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 1.0f) * u1)) * (((((0.6666666666666666f * (u2 * u2)) * ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI)))) + ((((float) M_PI) * ((float) M_PI)) * -2.0f)) * (u2 * u2)) + 1.0f);
	} else {
		tmp = sqrtf((((0.5f * u1) + 1.0f) * u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.04500000178813934))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))) * Float32(u2 * u2)) + Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(0.5) * u1) + Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.04500000178813934))
		tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * (((((single(0.6666666666666666) * (u2 * u2)) * ((single(pi) * single(pi)) * (single(pi) * single(pi)))) + ((single(pi) * single(pi)) * single(-2.0))) * (u2 * u2)) + single(1.0));
	else
		tmp = sqrt((((single(0.5) * u1) + single(1.0)) * u1)) * cos(((single(pi) + single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0450000018

   1. Initial program 62.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. sin-+PI/2-rev N/A

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

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

   8. Applied rewrites 94.5%

      \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      3. sqr-pow N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      7. pow2 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      12. lift-*.f32 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      14. lift-PI.f32 94.5

          \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   10. Applied rewrites 94.5%

       \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   if 0.0450000018 < u2

   1. Initial program 58.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 90.3

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

   5. Applied rewrites 90.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 90.3

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

   7. Applied rewrites 90.3%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 84.3%

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

Alternative 6: 91.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.11500000208616257:\\ \;\;\;\;\sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.11500000208616257)
   (*
    (sqrt
     (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1))
    (+
     (*
      (+
       (* (* 0.6666666666666666 (* u2 u2)) (* (* PI PI) (* PI PI)))
       (* (* PI PI) -2.0))
      (* u2 u2))
     1.0))
   (* (sqrt u1) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.11500000208616257f) {
		tmp = sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 1.0f) * u1)) * (((((0.6666666666666666f * (u2 * u2)) * ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI)))) + ((((float) M_PI) * ((float) M_PI)) * -2.0f)) * (u2 * u2)) + 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.11500000208616257))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))) * Float32(u2 * u2)) + Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.11500000208616257))
		tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * (((((single(0.6666666666666666) * (u2 * u2)) * ((single(pi) * single(pi)) * (single(pi) * single(pi)))) + ((single(pi) * single(pi)) * single(-2.0))) * (u2 * u2)) + single(1.0));
	else
		tmp = sqrt(u1) * cos(((single(pi) + single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.115000002

   1. Initial program 62.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. sin-+PI/2-rev N/A

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

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

   8. Applied rewrites 93.8%

      \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      3. sqr-pow N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      7. pow2 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      11. pow2 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      12. lift-*.f32 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

      14. lift-PI.f32 93.8

          \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   10. Applied rewrites 93.8%

       \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   if 0.115000002 < u2

   1. Initial program 59.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 90.2

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

   5. Applied rewrites 90.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 90.2

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. neg-log 73.2

         \[\leadsto \sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      2. flip3-- 73.2

         \[\leadsto \sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      3. metadata-eval 73.2

         \[\leadsto \sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. metadata-eval 73.2

         \[\leadsto \sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. neg-log 73.2

         \[\leadsto \sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

   10. Applied rewrites 73.2%

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

Alternative 7: 86.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1))
  (+
   (*
    (+
     (* (* 0.6666666666666666 (* u2 u2)) (* (* PI PI) (* PI PI)))
     (* (* PI PI) -2.0))
    (* u2 u2))
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 1.0f) * u1)) * (((((0.6666666666666666f * (u2 * u2)) * ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI)))) + ((((float) M_PI) * ((float) M_PI)) * -2.0f)) * (u2 * u2)) + 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * Float32(Float32(Float32(Float32(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))) * Float32(u2 * u2)) + Float32(1.0)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * (((((single(0.6666666666666666) * (u2 * u2)) * ((single(pi) * single(pi)) * (single(pi) * single(pi)))) + ((single(pi) * single(pi)) * single(-2.0))) * (u2 * u2)) + single(1.0));
end

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 93.8

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

5. Applied rewrites 93.8%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
7. Step-by-step derivation

   1. sin-+PI/2-rev N/A

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

   2. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

8. Applied rewrites 85.3%

   \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right)} \]
9. Step-by-step derivation

   1. lift-PI.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   2. lift-pow.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   3. sqr-pow N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{4}{2}\right)}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   7. pow2 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   9. lift-PI.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   10. lift-PI.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   11. pow2 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   13. lift-PI.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\frac{2}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

   14. lift-PI.f32 85.3

       \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

10. Applied rewrites 85.3%

    \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]
11. Add Preprocessing

Alternative 8: 83.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * ((((single(pi) * single(pi)) * single(-2.0)) * (u2 * u2)) + single(1.0));
end

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 93.8

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

5. Applied rewrites 93.8%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
7. Step-by-step derivation

   1. sin-+PI/2-rev N/A

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

   2. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \]

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

8. Applied rewrites 85.3%

   \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \color{blue}{\left(\left(\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{4} + \left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right)} \]

9. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

    2. pow2 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-PI.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]

    5. lift-PI.f32 N/A

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

    6. lift-*.f32 82.3

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

11. Applied rewrites 82.3%

    \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot -2\right) \cdot \left(u2 \cdot u2\right) + 1\right) \]
12. Add Preprocessing

Alternative 9: 76.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((((((((0.25e0 * u1) + 0.3333333333333333e0) * u1) + 0.5e0) * u1) + 1.0e0) * u1))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1));
end

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   4. lift-log.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   5. lift--.f32 52.0

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

5. Applied rewrites 52.0%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right) \cdot u1} \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1 + 1\right) \cdot u1} \]

   4. +-commutative N/A

      \[\leadsto \sqrt{\left(\left(u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   5. +-commutative N/A

      \[\leadsto \sqrt{\left(\left(u1 \cdot \left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   7. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   8. lift-+.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   9. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   10. lift-+.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   11. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   12. lift-+.f32 N/A

       \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \]

   13. lift-*.f32 75.6

       \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \]

8. Applied rewrites 75.6%

   \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \]

9. Final simplification 75.6%

   \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 10: 75.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 + u1 \cdot \left(\left(0.5 - -0.3333333333333333 \cdot u1\right) \cdot u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (+ u1 (* u1 (* (- 0.5 (* -0.3333333333333333 u1)) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 + (u1 * ((0.5f - (-0.3333333333333333f * u1)) * u1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 + (u1 * ((0.5e0 - ((-0.3333333333333333e0) * u1)) * u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 + Float32(u1 * Float32(Float32(Float32(0.5) - Float32(Float32(-0.3333333333333333) * u1)) * u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 + (u1 * ((single(0.5) - (single(-0.3333333333333333) * u1)) * u1))));
end

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   4. lift-log.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   5. lift--.f32 52.0

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

5. Applied rewrites 52.0%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
7. Step-by-step derivation

8. Applied rewrites 61.7%

   \[\leadsto \sqrt{u1} \]

9. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)} \]
10. Step-by-step derivation

    1. distribute-lft-in N/A

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

    2. *-commutative N/A

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

    3. *-lft-identity N/A

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

    4. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{u1 + u1 \cdot \left(u1 \cdot \left(\frac{1}{2} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot u1\right)\right)} \]

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

    7. lower-+.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. lift--.f32 N/A

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

    11. lift-*.f32 74.2

        \[\leadsto \sqrt{u1 + u1 \cdot \left(\left(0.5 - -0.3333333333333333 \cdot u1\right) \cdot u1\right)} \]

11. Applied rewrites 74.2%

    \[\leadsto \sqrt{u1 + u1 \cdot \left(\left(0.5 - -0.3333333333333333 \cdot u1\right) \cdot u1\right)} \]

12. Final simplification 74.2%

    \[\leadsto \sqrt{u1 + u1 \cdot \left(\left(0.5 - -0.3333333333333333 \cdot u1\right) \cdot u1\right)} \]
13. Add Preprocessing

Alternative 11: 75.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(0.5 - -0.3333333333333333 \cdot u1\right) \cdot u1 + 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (+ (* (- 0.5 (* -0.3333333333333333 u1)) u1) 1.0) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((((0.5f - (-0.3333333333333333f * u1)) * u1) + 1.0f) * u1));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((((0.5e0 - ((-0.3333333333333333e0) * u1)) * u1) + 1.0e0) * u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(Float32(Float32(Float32(0.5) - Float32(Float32(-0.3333333333333333) * u1)) * u1) + Float32(1.0)) * u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((((single(0.5) - (single(-0.3333333333333333) * u1)) * u1) + single(1.0)) * u1));
end

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   4. lift-log.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   5. lift--.f32 52.0

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

5. Applied rewrites 52.0%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. lift-*.f32 74.2

      \[\leadsto \sqrt{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \]

   10. lift-+.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. +-commutative N/A

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

   13. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{\left(\left(\frac{1}{2} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot u1\right) \cdot u1 + 1\right) \cdot u1} \]

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 74.2

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

8. Applied rewrites 74.2%

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

9. Final simplification 74.2%

   \[\leadsto \sqrt{\left(\left(0.5 - -0.3333333333333333 \cdot u1\right) \cdot u1 + 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 12: 72.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((((0.5e0 * u1) + 1.0e0) * u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   4. lift-log.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   5. lift--.f32 52.0

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

5. Applied rewrites 52.0%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot u1} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(1 + \frac{1}{2} \cdot u1\right) \cdot u1} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \]

   4. lower-+.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \]

   5. lower-*.f32 71.3

      \[\leadsto \sqrt{\left(0.5 \cdot u1 + 1\right) \cdot u1} \]

8. Applied rewrites 71.3%

   \[\leadsto \sqrt{\left(0.5 \cdot u1 + 1\right) \cdot u1} \]

9. Final simplification 71.3%

   \[\leadsto \sqrt{\left(0.5 \cdot u1 + 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 13: 64.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 62.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{-1}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   4. lift-log.f32 N/A

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

   5. lift--.f32 52.0

      \[\leadsto \sqrt{\log \left(1 - u1\right) \cdot -1} \]

5. Applied rewrites 52.0%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
7. Step-by-step derivation

8. Applied rewrites 61.7%

   \[\leadsto \sqrt{u1} \]

9. Final simplification 61.7%

   \[\leadsto \sqrt{u1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))