Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 15.1s

Alternatives: 21

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - u1}{u1}\\ \sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{\left(1 - u1\right) - t\_0 \cdot 0}{t\_0 \cdot \left(1 - u1\right)}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ (- 1.0 u1) u1)))
   (*
    (sin (* u2 6.28318530718))
    (sqrt (/ (- (- 1.0 u1) (* t_0 0.0)) (* t_0 (- 1.0 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f - u1) / u1;
	return sinf((u2 * 6.28318530718f)) * sqrtf((((1.0f - u1) - (t_0 * 0.0f)) / (t_0 * (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
    real(4) :: t_0
    t_0 = (1.0e0 - u1) / u1
    code = sin((u2 * 6.28318530718e0)) * sqrt((((1.0e0 - u1) - (t_0 * 0.0e0)) / (t_0 * (1.0e0 - u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) - u1) / u1)
	return Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(Float32(Float32(Float32(1.0) - u1) - Float32(t_0 * Float32(0.0))) / Float32(t_0 * Float32(Float32(1.0) - u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = (single(1.0) - u1) / u1;
	tmp = sin((u2 * single(6.28318530718))) * sqrt((((single(1.0) - u1) - (t_0 * single(0.0))) / (t_0 * (single(1.0) - u1))));
end

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Final simplification 98.6%

   \[\leadsto \sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{\left(1 - u1\right) - \frac{1 - u1}{u1} \cdot 0}{\frac{1 - u1}{u1} \cdot \left(1 - u1\right)}} \]
6. Add Preprocessing

Alternative 2: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.25:\\ \;\;\;\;\left(\mathsf{fma}\left(u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot u2, 81.6052492761019, -41.341702240407926 \cdot u2\right), 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.25)
   (*
    (*
     (fma
      u2
      (fma (* (* u2 u2) u2) 81.6052492761019 (* -41.341702240407926 u2))
      6.28318530718)
     u2)
    (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (fma (fma u1 u1 u1) u1 u1)) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.25f) {
		tmp = (fmaf(u2, fmaf(((u2 * u2) * u2), 81.6052492761019f, (-41.341702240407926f * u2)), 6.28318530718f) * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1)) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.25))
		tmp = Float32(Float32(fma(u2, fma(Float32(Float32(u2 * u2) * u2), Float32(81.6052492761019), Float32(Float32(-41.341702240407926) * u2)), Float32(6.28318530718)) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(fma(fma(u1, u1, u1), u1, u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.25

   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

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.6%

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

   if 0.25 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   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 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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.25:\\ \;\;\;\;\left(\mathsf{fma}\left(u2, \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot u2, 81.6052492761019, -41.341702240407926 \cdot u2\right), 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.800000011920929:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right), \left(\left(\mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right) \cdot t\_0\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(u2 \cdot u2\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* u2 6.28318530718) 0.800000011920929)
     (*
      (fma
       t_0
       (fma -41.341702240407926 (* u2 u2) 6.28318530718)
       (*
        (*
         (* (fma (* u2 u2) -76.70585975309672 81.6052492761019) t_0)
         (* u2 u2))
        (* u2 u2)))
      u2)
     (* (sqrt (fma u1 u1 u1)) (sin (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.800000011920929f) {
		tmp = fmaf(t_0, fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f), (((fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f) * t_0) * (u2 * u2)) * (u2 * u2))) * u2;
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * sinf((u2 * 6.28318530718f));
	}
	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(u2 * Float32(6.28318530718)) <= Float32(0.800000011920929))
		tmp = Float32(fma(t_0, fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)), Float32(Float32(Float32(fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)) * t_0) * Float32(u2 * u2)) * Float32(u2 * u2))) * u2);
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.800000012

   1. Initial program 98.5%

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

      1. lift-sqrt.f32 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 98.4%

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

   if 0.800000012 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. lower-fma.f32 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.800000011920929:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}}, \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right), \left(\left(\mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(u2 \cdot u2\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Final simplification 98.5%

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

Alternative 5: 93.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

   1. lift-sqrt.f32 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. lower-sqrt.f32 97.9

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

4. Applied rewrites 97.9%

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

5. Taylor expanded in u2 around 0

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

7. Applied rewrites 94.2%

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

8. Final simplification 94.2%

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

Alternative 6: 93.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Applied rewrites 94.2%

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

6. Final simplification 94.2%

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

Alternative 7: 85.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right) \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\left(1 - u1\right) \cdot u1}} \cdot \left(\left(6.28318530718 \cdot u1\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 1.500000053056283e-6)
   (*
    (*
     (sqrt u1)
     (fma
      (* (fma 81.6052492761019 (* u2 u2) -41.341702240407926) u2)
      u2
      6.28318530718))
    u2)
   (* (sqrt (/ 1.0 (* (- 1.0 u1) u1))) (* (* 6.28318530718 u1) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 1.500000053056283e-6f) {
		tmp = (sqrtf(u1) * fmaf((fmaf(81.6052492761019f, (u2 * u2), -41.341702240407926f) * u2), u2, 6.28318530718f)) * u2;
	} else {
		tmp = sqrtf((1.0f / ((1.0f - u1) * u1))) * ((6.28318530718f * u1) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(1.500000053056283e-6))
		tmp = Float32(Float32(sqrt(u1) * fma(Float32(fma(Float32(81.6052492761019), Float32(u2 * u2), Float32(-41.341702240407926)) * u2), u2, Float32(6.28318530718))) * u2);
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - u1) * u1))) * Float32(Float32(Float32(6.28318530718) * u1) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 1.50000005e-6

   1. Initial program 98.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. Applied rewrites 90.7%

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot u2 + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{3}\right)\right)\right) \cdot u2 \]
   9. Step-by-step derivation

   10. Applied rewrites 90.1%

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

   if 1.50000005e-6 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.3%

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

   4. Applied rewrites 98.5%

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \frac{u1 - 1}{u1}} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. mul0-lft N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. neg-sub0 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. neg-mul-1 N/A

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

      9. times-frac N/A

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

   6. Applied rewrites 98.2%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-lft-identity N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f32 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-*.f32 N/A

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

       11. lower-sqrt.f32 N/A

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

       12. lower-/.f32 N/A

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

       13. sub-neg N/A

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

       14. *-rgt-identity N/A

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

       15. unpow2 N/A

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

       16. distribute-rgt-neg-in N/A

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

       17. mul-1-neg N/A

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

       18. distribute-lft-out N/A

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

       19. lower-*.f32 N/A

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

       20. mul-1-neg N/A

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

       21. sub-neg N/A

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

       22. lower--.f32 83.4

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

   11. Applied rewrites 83.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right) \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\left(1 - u1\right) \cdot u1}} \cdot \left(\left(6.28318530718 \cdot u1\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right) \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot u2\right) \cdot 6.28318530718\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 1.500000053056283e-6)
     (*
      (*
       (sqrt u1)
       (fma
        (* (fma 81.6052492761019 (* u2 u2) -41.341702240407926) u2)
        u2
        6.28318530718))
      u2)
     (* (* (sqrt t_0) u2) 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 1.500000053056283e-6f) {
		tmp = (sqrtf(u1) * fmaf((fmaf(81.6052492761019f, (u2 * u2), -41.341702240407926f) * u2), u2, 6.28318530718f)) * u2;
	} else {
		tmp = (sqrtf(t_0) * u2) * 6.28318530718f;
	}
	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(1.500000053056283e-6))
		tmp = Float32(Float32(sqrt(u1) * fma(Float32(fma(Float32(81.6052492761019), Float32(u2 * u2), Float32(-41.341702240407926)) * u2), u2, Float32(6.28318530718))) * u2);
	else
		tmp = Float32(Float32(sqrt(t_0) * u2) * Float32(6.28318530718));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 1.50000005e-6

   1. Initial program 98.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. Applied rewrites 90.7%

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot u2 + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{3}\right)\right)\right) \cdot u2 \]
   9. Step-by-step derivation

   10. Applied rewrites 90.1%

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

   if 1.50000005e-6 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.3%

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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right) \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right) \cdot 6.28318530718\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right) \cdot u2, u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot u2\right) \cdot 6.28318530718\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 1.500000053056283e-6)
     (*
      (*
       (fma
        (* (fma 81.6052492761019 (* u2 u2) -41.341702240407926) u2)
        u2
        6.28318530718)
       u2)
      (sqrt u1))
     (* (* (sqrt t_0) u2) 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 1.500000053056283e-6f) {
		tmp = (fmaf((fmaf(81.6052492761019f, (u2 * u2), -41.341702240407926f) * u2), u2, 6.28318530718f) * u2) * sqrtf(u1);
	} else {
		tmp = (sqrtf(t_0) * u2) * 6.28318530718f;
	}
	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(1.500000053056283e-6))
		tmp = Float32(Float32(fma(Float32(fma(Float32(81.6052492761019), Float32(u2 * u2), Float32(-41.341702240407926)) * u2), u2, Float32(6.28318530718)) * u2) * sqrt(u1));
	else
		tmp = Float32(Float32(sqrt(t_0) * u2) * Float32(6.28318530718));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 1.50000005e-6

   1. Initial program 98.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. Applied rewrites 90.7%

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot u2 + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{3}\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 90.0%

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

   if 1.50000005e-6 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.3%

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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

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

Alternative 10: 91.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

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

5. Applied rewrites 92.1%

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

7. Applied rewrites 92.2%

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

Alternative 11: 91.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

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

5. Applied rewrites 92.1%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 92.1%

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

9. Final simplification 92.1%

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

Alternative 12: 84.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot u2\right) \cdot 6.28318530718\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 1.500000053056283e-6)
     (* (* (sqrt u1) (fma (* -41.341702240407926 u2) u2 6.28318530718)) u2)
     (* (* (sqrt t_0) u2) 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 1.500000053056283e-6f) {
		tmp = (sqrtf(u1) * fmaf((-41.341702240407926f * u2), u2, 6.28318530718f)) * u2;
	} else {
		tmp = (sqrtf(t_0) * u2) * 6.28318530718f;
	}
	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(1.500000053056283e-6))
		tmp = Float32(Float32(sqrt(u1) * fma(Float32(Float32(-41.341702240407926) * u2), u2, Float32(6.28318530718))) * u2);
	else
		tmp = Float32(Float32(sqrt(t_0) * u2) * Float32(6.28318530718));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 1.50000005e-6

   1. Initial program 98.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. Applied rewrites 86.6%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 86.1%

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

   if 1.50000005e-6 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.3%

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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.3%

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

Alternative 13: 89.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f32 85.4

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

7. Applied rewrites 85.4%

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

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. associate-*r* N/A

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

   9. unpow2 N/A

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

   10. unpow3 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{314159265359}{50000000000} \cdot u2\right) + \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \color{blue}{{u2}^{3}} \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{314159265359}{50000000000} \cdot u2\right) + \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)} \cdot {u2}^{3} \]

   12. associate-*l* N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{314159265359}{50000000000} \cdot u2\right) + \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{3}\right)} \]

   13. unpow3 N/A

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

   14. unpow2 N/A

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

10. Applied rewrites 89.5%

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

11. Final simplification 89.5%

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

Alternative 14: 88.9% accurate, 2.9× speedup

Math

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

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(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2) * fma(Float32(Float32(-41.341702240407926) * u2), u2, Float32(6.28318530718)))
end

Derivation

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

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

5. Applied rewrites 89.3%

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

7. Applied rewrites 89.4%

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

8. Final simplification 89.4%

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

Alternative 15: 89.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

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

5. Applied rewrites 89.3%

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

6. Final simplification 89.3%

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

Alternative 16: 78.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.02500000037252903)
   (* (sqrt (fma (fma u1 u1 u1) u1 u1)) (* u2 6.28318530718))
   (* (* (sqrt u1) (fma (* -41.341702240407926 u2) u2 6.28318530718)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.02500000037252903f) {
		tmp = sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1)) * (u2 * 6.28318530718f);
	} else {
		tmp = (sqrtf(u1) * fmaf((-41.341702240407926f * u2), u2, 6.28318530718f)) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.02500000037252903))
		tmp = Float32(sqrt(fma(fma(u1, u1, u1), u1, u1)) * Float32(u2 * Float32(6.28318530718)));
	else
		tmp = Float32(Float32(sqrt(u1) * fma(Float32(Float32(-41.341702240407926) * u2), u2, Float32(6.28318530718))) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0250000004

   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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 87.0%

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

   if 0.0250000004 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. Applied rewrites 63.7%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 52.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.00800000037997961)
   (* (sqrt (fma u1 u1 u1)) (* u2 6.28318530718))
   (* (* (sqrt u1) (fma (* -41.341702240407926 u2) u2 6.28318530718)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.00800000037997961f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * (u2 * 6.28318530718f);
	} else {
		tmp = (sqrtf(u1) * fmaf((-41.341702240407926f * u2), u2, 6.28318530718f)) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.00800000037997961))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(u2 * Float32(6.28318530718)));
	else
		tmp = Float32(Float32(sqrt(u1) * fma(Float32(Float32(-41.341702240407926) * u2), u2, Float32(6.28318530718))) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00800000038

   1. Initial program 98.6%

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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 84.2%

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

   if 0.00800000038 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. Applied rewrites 69.7%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 55.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{u1} \cdot \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right)\right) \cdot u2\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.00800000037997961)
   (* (sqrt (fma u1 u1 u1)) (* u2 6.28318530718))
   (* (sqrt u1) (* (fma (* u2 u2) -41.341702240407926 6.28318530718) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.00800000037997961f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * (u2 * 6.28318530718f);
	} else {
		tmp = sqrtf(u1) * (fmaf((u2 * u2), -41.341702240407926f, 6.28318530718f) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.00800000037997961))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(u2 * Float32(6.28318530718)));
	else
		tmp = Float32(sqrt(u1) * Float32(fma(Float32(u2 * u2), Float32(-41.341702240407926), Float32(6.28318530718)) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00800000038

   1. Initial program 98.6%

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

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 84.2%

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

   if 0.00800000038 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. Applied rewrites 69.7%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 81.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

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))) * (u2 * 6.28318530718e0)
end function

Julia

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

MATLAB

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

Derivation

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 81.3%

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

Alternative 20: 73.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 81.3%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 71.8%

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

Alternative 21: 64.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

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) * u2) * 6.28318530718e0
end function

Julia

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

MATLAB

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

Derivation

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 81.3%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 63.5%

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

9. Final simplification 63.5%

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

Reproduce

herbie shell --seed 2024235 
(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))))