Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 16.1s

Alternatives: 23

Speedup: 1.0×

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

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Initial Program: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Alternative 1: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ (sin (* 6.28318530718 u2)) (sqrt (+ (/ 1.0 u1) -1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) / sqrtf(((1.0f / u1) + -1.0f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((6.28318530718e0 * u2)) / sqrt(((1.0e0 / u1) + (-1.0e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) / sqrt(((single(1.0) / u1) + single(-1.0)));
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   2. clear-num N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   3. sqrt-div N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

   4. metadata-eval N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   7. sin-lowering-sin.f32 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   10. div-sub N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

   11. sub-neg N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)}}} \]

   12. *-inverses N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + -1}}} \]

   15. /-lowering-/.f32 98.2

       \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1}} + -1}} \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1} + -1}}} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 1.399999976158142:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, t\_0, \left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 1.399999976158142)
     (*
      u2
      (fma
       6.28318530718
       t_0
       (*
        (* u2 u2)
        (*
         t_0
         (fma
          (* u2 u2)
          (fma u2 (* u2 -76.70585975309672) 81.6052492761019)
          -41.341702240407926)))))
     (* (sin (* 6.28318530718 u2)) (sqrt (fma u1 (fma u1 u1 u1) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 1.399999976158142f) {
		tmp = u2 * fmaf(6.28318530718f, t_0, ((u2 * u2) * (t_0 * fmaf((u2 * u2), fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f), -41.341702240407926f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(1.399999976158142))
		tmp = Float32(u2 * fma(Float32(6.28318530718), t_0, Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.39999998

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)} \]

   if 1.39999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(1 + u1\right) + 1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + u1 \cdot 1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + \color{blue}{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(1 + u1\right), u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \color{blue}{\left(u1 + 1\right)}, u1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot u1 + u1 \cdot 1}, u1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. accelerator-lowering-fma.f32 89.3

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. Simplified 89.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 1.399999976158142:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 1.399999976158142:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, t\_0, \left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 1.399999976158142)
     (*
      u2
      (fma
       6.28318530718
       t_0
       (*
        (* u2 u2)
        (*
         t_0
         (fma
          (* u2 u2)
          (fma u2 (* u2 -76.70585975309672) 81.6052492761019)
          -41.341702240407926)))))
     (* (sin (* 6.28318530718 u2)) (sqrt (fma u1 u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 1.399999976158142f) {
		tmp = u2 * fmaf(6.28318530718f, t_0, ((u2 * u2) * (t_0 * fmaf((u2 * u2), fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f), -41.341702240407926f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(fmaf(u1, u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(1.399999976158142))
		tmp = Float32(u2 * fma(Float32(6.28318530718), t_0, Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.39999998

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)} \]

   if 1.39999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot u1 + u1 \cdot 1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot u1 + \color{blue}{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. accelerator-lowering-fma.f32 87.8

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. Simplified 87.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 1.399999976158142:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 6.28318530718 u2)) (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf((u1 / (1.0f - u1)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 / (single(1.0) - u1)));
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Final simplification 98.2%

   \[\leadsto \sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
4. Add Preprocessing

Alternative 5: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 1.399999976158142:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, t\_0, \left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 1.399999976158142)
     (*
      u2
      (fma
       6.28318530718
       t_0
       (*
        (* u2 u2)
        (*
         t_0
         (fma
          (* u2 u2)
          (fma u2 (* u2 -76.70585975309672) 81.6052492761019)
          -41.341702240407926)))))
     (* (sin (* 6.28318530718 u2)) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 1.399999976158142f) {
		tmp = u2 * fmaf(6.28318530718f, t_0, ((u2 * u2) * (t_0 * fmaf((u2 * u2), fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f), -41.341702240407926f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(1.399999976158142))
		tmp = Float32(u2 * fma(Float32(6.28318530718), t_0, Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.39999998

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)} \]

   if 1.39999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. sin-lowering-sin.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      4. *-lowering-*.f32 79.7

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

   5. Simplified 79.7%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 1.399999976158142:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ u2 \cdot \mathsf{fma}\left(6.28318530718, t\_0, \left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    u2
    (fma
     6.28318530718
     t_0
     (*
      (* u2 u2)
      (*
       t_0
       (fma
        (* u2 u2)
        (fma u2 (* u2 -76.70585975309672) 81.6052492761019)
        -41.341702240407926)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return u2 * fmaf(6.28318530718f, t_0, ((u2 * u2) * (t_0 * fmaf((u2 * u2), fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f), -41.341702240407926f))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(u2 * fma(Float32(6.28318530718), t_0, Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926))))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Simplified 92.8%

   \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)} \]
6. Add Preprocessing

Alternative 7: 93.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/
  (*
   u2
   (fma
    (* u2 u2)
    (fma
     (* u2 u2)
     (fma u2 (* u2 -76.70585975309672) 81.6052492761019)
     -41.341702240407926)
    6.28318530718))
  (sqrt (+ (/ 1.0 u1) -1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * fmaf((u2 * u2), fmaf((u2 * u2), fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f), -41.341702240407926f), 6.28318530718f)) / sqrtf(((1.0f / u1) + -1.0f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   2. clear-num N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   3. sqrt-div N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

   4. metadata-eval N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   7. sin-lowering-sin.f32 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   10. div-sub N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

   11. sub-neg N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)}}} \]

   12. *-inverses N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + -1}}} \]

   15. /-lowering-/.f32 98.2

       \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1}} + -1}} \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1} + -1}}} \]

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   4. unpow2 N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   6. sub-neg N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   9. unpow2 N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   11. +-commutative N/A

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   12. *-commutative N/A

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   13. unpow2 N/A

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   14. associate-*l* N/A

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{u2 \cdot \left(u2 \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   15. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   16. *-lowering-*.f32 92.8

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, \color{blue}{u2 \cdot -76.70585975309672}, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \]

7. Simplified 92.8%

   \[\leadsto \frac{\color{blue}{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
8. Add Preprocessing

Alternative 8: 93.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (*
   u2
   (fma
    (* u2 u2)
    (fma
     (* u2 u2)
     (fma u2 (* u2 -76.70585975309672) 81.6052492761019)
     -41.341702240407926)
    6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * fmaf((u2 * u2), fmaf((u2 * u2), fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f), -41.341702240407926f), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{u2 \cdot \left(u2 \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   15. accelerator-lowering-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   16. *-lowering-*.f32 92.8

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, \color{blue}{u2 \cdot -76.70585975309672}, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

5. Simplified 92.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)} \]
6. Add Preprocessing

Alternative 9: 87.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.0044999998062849045:\\ \;\;\;\;u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.0044999998062849045)
     (*
      u2
      (*
       (sqrt (fma u1 (fma u1 u1 u1) u1))
       (fma -41.341702240407926 (* u2 u2) 6.28318530718)))
     (* 6.28318530718 (* u2 (sqrt t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.0044999998062849045f) {
		tmp = u2 * (sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1)) * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	} else {
		tmp = 6.28318530718f * (u2 * sqrtf(t_0));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0044999998062849045))
		tmp = Float32(u2 * Float32(sqrt(fma(u1, fma(u1, u1, u1), u1)) * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00449999981

   1. Initial program 98.2%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(1 + u1\right) + 1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + u1 \cdot 1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + \color{blue}{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(1 + u1\right), u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \color{blue}{\left(u1 + 1\right)}, u1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot u1 + u1 \cdot 1}, u1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. accelerator-lowering-fma.f32 98.2

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. Simplified 98.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto u2 \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)} + \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}}\right) \]

      3. *-commutative N/A

         \[\leadsto u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)} + \color{blue}{\left({u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)} + \color{blue}{{u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}\right)}\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)} + \color{blue}{\left({u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right) \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}}\right) \]

      7. *-commutative N/A

         \[\leadsto u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)} + \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)} \cdot \sqrt{u1 + u1 \cdot \left(u1 + {u1}^{2}\right)}\right) \]

   8. Simplified 87.7%

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   if 0.00449999981 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.2%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. /-lowering-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \frac{314159265359}{50000000000}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \frac{314159265359}{50000000000}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \cdot \frac{314159265359}{50000000000} \]

      5. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \left(u2 \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}}\right) \cdot \frac{314159265359}{50000000000} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \left(u2 \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right) \cdot \frac{314159265359}{50000000000} \]

      7. --lowering--.f32 83.3

         \[\leadsto \left(u2 \cdot \sqrt{\frac{u1}{\color{blue}{1 - u1}}}\right) \cdot 6.28318530718 \]

   7. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot 6.28318530718} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0044999998062849045:\\ \;\;\;\;u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/
  (*
   u2
   (fma
    (* u2 u2)
    (fma (* u2 u2) 81.6052492761019 -41.341702240407926)
    6.28318530718))
  (sqrt (+ (/ 1.0 u1) -1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * fmaf((u2 * u2), fmaf((u2 * u2), 81.6052492761019f, -41.341702240407926f), 6.28318530718f)) / sqrtf(((1.0f / u1) + -1.0f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(81.6052492761019), Float32(-41.341702240407926)), Float32(6.28318530718))) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   2. clear-num N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   3. sqrt-div N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

   4. metadata-eval N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   7. sin-lowering-sin.f32 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   9. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   10. div-sub N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

   11. sub-neg N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)}}} \]

   12. *-inverses N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + -1}}} \]

   15. /-lowering-/.f32 98.2

       \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1}} + -1}} \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1} + -1}}} \]

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   2. +-commutative N/A

      \[\leadsto \frac{u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   4. unpow2 N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   6. sub-neg N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   7. *-commutative N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   10. unpow2 N/A

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   11. *-lowering-*.f32 90.6

       \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \]

7. Simplified 90.6%

   \[\leadsto \frac{\color{blue}{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
8. Add Preprocessing

Alternative 11: 86.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.0008200000156648457)
     (*
      (sqrt (fma u1 u1 u1))
      (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)))
     (* 6.28318530718 (* u2 (sqrt t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.0008200000156648457f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	} else {
		tmp = 6.28318530718f * (u2 * sqrtf(t_0));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0008200000156648457))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 8.20000016e-4

   1. Initial program 98.2%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   5. Simplified 87.5%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

      3. distribute-lft1-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot u1 + u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, u2 \cdot u2, \frac{314159265359}{50000000000}\right)\right) \]

      4. accelerator-lowering-fma.f32 87.3

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \]

   8. Simplified 87.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \]

   if 8.20000016e-4 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.1%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. /-lowering-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 83.9%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \frac{314159265359}{50000000000}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \frac{314159265359}{50000000000}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \cdot \frac{314159265359}{50000000000} \]

      5. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \left(u2 \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}}\right) \cdot \frac{314159265359}{50000000000} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \left(u2 \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right) \cdot \frac{314159265359}{50000000000} \]

      7. --lowering--.f32 84.1

         \[\leadsto \left(u2 \cdot \sqrt{\frac{u1}{\color{blue}{1 - u1}}}\right) \cdot 6.28318530718 \]

   7. Applied egg-rr 84.1%

      \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot 6.28318530718} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (*
   u2
   (fma
    u2
    (* u2 (fma u2 (* u2 81.6052492761019) -41.341702240407926))
    6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * fmaf(u2, (u2 * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f)), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * fma(u2, Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(u2 \cdot \color{blue}{\left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) \cdot u2\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) \cdot u2, \frac{314159265359}{50000000000}\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot u2\right) \cdot u2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot u2\right)} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot u2\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

   14. accelerator-lowering-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left(u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   16. *-lowering-*.f32 90.6

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot 81.6052492761019}, -41.341702240407926\right), 6.28318530718\right)\right) \]

5. Simplified 90.6%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]
6. Add Preprocessing

Alternative 13: 91.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (sqrt (/ u1 (- 1.0 u1)))
   (fma
    u2
    (* u2 (fma u2 (* u2 81.6052492761019) -41.341702240407926))
    6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf((u1 / (1.0f - u1))) * fmaf(u2, (u2 * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f)), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(u2, Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)\right)\right)} \]

5. Simplified 90.6%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]
6. Add Preprocessing

Alternative 14: 89.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, 6.28318530718, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma u2 6.28318530718 (* u2 (* (* u2 u2) -41.341702240407926)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf(u2, 6.28318530718f, (u2 * ((u2 * u2) * -41.341702240407926f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(u2, Float32(6.28318530718), Float32(u2 * Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

5. Simplified 87.8%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000} + u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)\right)} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(u2, \frac{314159265359}{50000000000}, u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, \frac{314159265359}{50000000000}, \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, \frac{314159265359}{50000000000}, u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(u2 \cdot u2\right)\right)}\right) \]

   6. *-lowering-*.f32 87.9

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, 6.28318530718, u2 \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

7. Applied egg-rr 87.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(u2, 6.28318530718, u2 \cdot \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

8. Final simplification 87.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, 6.28318530718, u2 \cdot \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \]
9. Add Preprocessing

Alternative 15: 84.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004000000189989805)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1)))))
   (* (sqrt u1) (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004000000189989805f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00400000019

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. /-lowering-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \frac{314159265359}{50000000000}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \frac{314159265359}{50000000000}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \cdot \frac{314159265359}{50000000000} \]

      5. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \left(u2 \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}}\right) \cdot \frac{314159265359}{50000000000} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \left(u2 \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right) \cdot \frac{314159265359}{50000000000} \]

      7. --lowering--.f32 97.7

         \[\leadsto \left(u2 \cdot \sqrt{\frac{u1}{\color{blue}{1 - u1}}}\right) \cdot 6.28318530718 \]

   7. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot 6.28318530718} \]

   if 0.00400000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

      7. *-lowering-*.f32 56.7

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   8. Simplified 56.7%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004000000189989805)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt u1) (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004000000189989805f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00400000019

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. /-lowering-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   if 0.00400000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

      7. *-lowering-*.f32 56.7

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   8. Simplified 56.7%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 89.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

5. Simplified 87.8%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
6. Add Preprocessing

Alternative 18: 79.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004000000189989805)
   (* (* 6.28318530718 u2) (sqrt (fma u1 (fma u1 u1 u1) u1)))
   (* (sqrt u1) (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004000000189989805f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00400000019

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. /-lowering-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(1 + u1\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + u1 \cdot 1}} \]

      3. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 \cdot \left(1 + u1\right)\right) + \color{blue}{u1}} \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(1 + u1\right), u1\right)}} \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \color{blue}{\left(1 + u1\right) \cdot u1}, u1\right)} \]

      6. +-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \color{blue}{\left(u1 + 1\right)} \cdot u1, u1\right)} \]

      7. distribute-lft1-in N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot u1 + u1}, u1\right)} \]

      8. accelerator-lowering-fma.f32 90.6

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)} \]

   8. Simplified 90.6%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}} \]

   if 0.00400000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

      7. *-lowering-*.f32 56.7

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   8. Simplified 56.7%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 76.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004000000189989805)
   (* (* 6.28318530718 u2) (sqrt (fma u1 u1 u1)))
   (* (sqrt u1) (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004000000189989805f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, u1, u1));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.004000000189989805))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, u1, u1)));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00400000019

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. /-lowering-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot u1 + u1}} \]

      4. accelerator-lowering-fma.f32 86.5

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

   8. Simplified 86.5%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

   if 0.00400000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

      7. *-lowering-*.f32 56.7

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   8. Simplified 56.7%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 73.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 6.28318530718 u2) (sqrt (fma u1 u1 u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * sqrtf(fmaf(u1, u1, u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, u1, u1)))
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

   6. sub-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   7. rgt-mult-inverse N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   8. mul-1-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   10. mul-1-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

   11. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

   12. distribute-lft-in N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   13. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   14. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

   15. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   16. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   17. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   19. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

   20. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   21. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

5. Simplified 80.9%

   \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \]

   3. distribute-lft1-in N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot u1 + u1}} \]

   4. accelerator-lowering-fma.f32 73.3

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

8. Simplified 73.3%

   \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]
9. Add Preprocessing

Alternative 21: 64.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 6.28318530718 u2) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (6.28318530718e0 * u2) * sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * sqrt(u1);
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

   6. sub-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   7. rgt-mult-inverse N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   8. mul-1-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   10. mul-1-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

   11. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

   12. distribute-lft-in N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   13. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   14. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

   15. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   16. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   17. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   19. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

   20. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   21. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

5. Simplified 80.9%

   \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{u1}} \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 65.2

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{\sqrt{u1}} \]

8. Simplified 65.2%

   \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{\sqrt{u1}} \]
9. Add Preprocessing

Alternative 22: 64.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf(u1));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt(u1));
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

   6. sub-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   7. rgt-mult-inverse N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   8. mul-1-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   10. mul-1-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

   11. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

   12. distribute-lft-in N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   13. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   14. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

   15. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   16. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   17. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   19. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

   20. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   21. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

5. Simplified 80.9%

   \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \]

   4. sqrt-lowering-sqrt.f32 65.1

      \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \color{blue}{\sqrt{u1}}\right) \]

8. Simplified 65.1%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right)} \]
9. Add Preprocessing

Alternative 23: 19.5% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot u2 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 6.28318530718 u2))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * u2;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * u2
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * u2)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * u2;
end

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 74.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{u1 + 1}}{\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}}}} \]

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{50000000000}{314159265359} \cdot \left(\sqrt{\frac{1 + u1}{u1}} \cdot \frac{1}{u2}\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{50000000000}{314159265359} \cdot \sqrt{\frac{1 + u1}{u1}}\right) \cdot \frac{1}{u2}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(\frac{50000000000}{314159265359} \cdot \sqrt{\frac{1 + u1}{u1}}\right) \cdot 1}{u2}}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{50000000000}{314159265359} \cdot \sqrt{\frac{1 + u1}{u1}}}}{u2}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{50000000000}{314159265359} \cdot \sqrt{\frac{1 + u1}{u1}}}{u2}}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{50000000000}{314159265359} \cdot \sqrt{\frac{1 + u1}{u1}}}}{u2}} \]

   6. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \frac{1}{\frac{\frac{50000000000}{314159265359} \cdot \color{blue}{\sqrt{\frac{1 + u1}{u1}}}}{u2}} \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \frac{1}{\frac{\frac{50000000000}{314159265359} \cdot \sqrt{\color{blue}{\frac{1 + u1}{u1}}}}{u2}} \]

   8. +-lowering-+.f32 63.5

      \[\leadsto \frac{1}{\frac{0.15915494309188485 \cdot \sqrt{\frac{\color{blue}{1 + u1}}{u1}}}{u2}} \]

7. Simplified 63.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{0.15915494309188485 \cdot \sqrt{\frac{1 + u1}{u1}}}{u2}}} \]

8. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot u2} \]
9. Step-by-step derivation

   1. *-lowering-*.f32 19.3

      \[\leadsto \color{blue}{6.28318530718 \cdot u2} \]

10. Simplified 19.3%

    \[\leadsto \color{blue}{6.28318530718 \cdot u2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))