Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.1s

Alternatives: 18

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, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (/ (- 1.0 u1) u1) -0.5) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f - u1) / u1), -0.5f) * 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 = (((1.0e0 - u1) / u1) ** (-0.5e0)) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) - u1) / u1) ^ Float32(-0.5)) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) - u1) / u1) ^ single(-0.5)) * sin((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 rewrites 98.3%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. sqrt-div N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

6. Applied rewrites 98.3%

   \[\leadsto \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{-0.5}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Add Preprocessing

Alternative 2: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 98.3%

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

5. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf((u1 / (((u1 * u1) + -1.0f) / (-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 / (((u1 * u1) + (-1.0e0)) / ((-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(Float32(u1 * u1) + Float32(-1.0)) / Float32(Float32(-1.0) - u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 / (((u1 * u1) + single(-1.0)) / (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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-neg.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Final simplification 98.3%

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

Alternative 4: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6000000238418579:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, t\_0, \left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.6000000238418579)
     (*
      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) <= 0.6000000238418579f) {
		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(0.6000000238418579))
		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) < 0.600000024

   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

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

   5. Applied rewrites 98.7%

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

   if 0.600000024 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 95.9%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   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. lower-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. lower-fma.f32 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6000000238418579:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.9800000190734863:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, t\_0, \left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.9800000190734863)
     (*
      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) <= 0.9800000190734863f) {
		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(0.9800000190734863))
		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) < 0.980000019

   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

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

   5. Applied rewrites 98.6%

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

   if 0.980000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 95.1%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.9800000190734863:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, \sqrt{\frac{u1}{1 - u1}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 7: 94.1% 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. Applied rewrites 93.2%

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

Alternative 8: 94.0% accurate, 2.0× speedup

Math

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

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(u2, Float32(u2 * fma(Float32(u2 * 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

4. Applied rewrites 98.3%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. sqrt-div N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 93.0

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

9. Applied rewrites 93.0%

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

11. Applied rewrites 93.0%

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

Alternative 9: 94.0% accurate, 2.0× speedup

Math

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

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(u2, Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * 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

4. Applied rewrites 98.3%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. sqrt-div N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 93.0

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

9. Applied rewrites 93.0%

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

11. Applied rewrites 93.0%

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

Alternative 10: 94.0% accurate, 2.0× speedup

Math

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

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(u2, Float32(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

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

   1. lower-*.f32 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u2 around 0

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

   1. lower-*.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. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f32 N/A

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

8. Applied rewrites 93.0%

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

Alternative 11: 91.9% 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. Applied rewrites 91.0%

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

Alternative 12: 86.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.03200000151991844:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.03200000151991844)
   (* (sqrt (/ u1 (- 1.0 u1))) (* 6.28318530718 u2))
   (*
    (sqrt u1)
    (*
     u2
     (fma
      (* u2 u2)
      (fma (* u2 u2) 81.6052492761019 -41.341702240407926)
      6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.03200000151991844f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (6.28318530718f * u2);
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((u2 * u2), fmaf((u2 * u2), 81.6052492761019f, -41.341702240407926f), 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.03200000151991844))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) * u2));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(81.6052492761019), Float32(-41.341702240407926)), Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0320000015

   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

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

      1. lower-*.f32 93.9

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

   5. Applied rewrites 93.9%

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

   if 0.0320000015 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.2%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-*.f32 37.8

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 36.6

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

   8. Applied rewrites 36.6%

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

   9. Taylor expanded in u2 around 0

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

       1. lower-*.f32 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. lower-fma.f32 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f32 55.2

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

   11. Applied rewrites 55.2%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 89.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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

   1. lower-*.f32 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. lower-*.f32 N/A

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

8. Applied rewrites 88.1%

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

Alternative 14: 81.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (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))) * (6.28318530718e0 * u2)
end function

Julia

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

MATLAB

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 80.1

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

5. Applied rewrites 80.1%

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

Alternative 15: 76.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f32 73.0

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

8. Applied rewrites 73.0%

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

Alternative 16: 73.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 69.9

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

8. Applied rewrites 69.9%

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

Alternative 17: 65.0% 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

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

   1. lower-*.f32 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 61.4

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

8. Applied rewrites 61.4%

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

9. Final simplification 61.4%

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

Alternative 18: 19.5% accurate, 22.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = u2 * 6.28318530718e0
end function

Julia

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

MATLAB

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

4. Applied rewrites 82.1%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. lower-*.f32 N/A

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

7. Applied rewrites 74.3%

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

8. Taylor expanded in u1 around inf

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

10. Applied rewrites 20.3%

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

11. Taylor expanded in u2 around 0

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

13. Applied rewrites 20.3%

    \[\leadsto u2 \cdot 6.28318530718 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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))))