Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.2%

Time: 22.4s

Alternatives: 8

Speedup: 1.0×

• unsoundness detected in the egraph

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (*
 (sqrt (/ u1 (- 1.0 u1)))
 (sin (fma -6.28318530718 u2 1.5707963705062866))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(fmaf(-6.28318530718f, u2, 1.5707963705062866f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(fma(Float32(-6.28318530718), u2, Float32(1.5707963705062866))))
end

Derivation

1. Initial program 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{-314159265359}{50000000000}}, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   9. mult-flip N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   13. lower-PI.f32 99.2%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \color{blue}{\pi}\right)\right) \]

3. Applied rewrites 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right)} \]

4. Evaluated real constant 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \color{blue}{1.5707963705062866}\right)\right) \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999988079071045:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \sin 1.5707963705062866\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (let* ((t_0 (cos (* 6.28318530718 u2))))
  (if (<= t_0 0.9999988079071045)
    (* (sqrt (fma u1 u1 u1)) t_0)
    (* (sqrt (/ u1 (- 1.0 u1))) (sin 1.5707963705062866)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999988079071045f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * t_0;
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * sinf(1.5707963705062866f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999988079071045))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(1.5707963705062866)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999998808

   1. Initial program 99.0%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 86.4%

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 86.5%

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

   6. Applied rewrites 86.5%

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

   if 0.999998808 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{-314159265359}{50000000000}}, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      13. lower-PI.f32 99.2%

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \color{blue}{\pi}\right)\right) \]

   3. Applied rewrites 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right)} \]

   4. Evaluated real constant 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \color{blue}{1.5707963705062866}\right)\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\frac{13176795}{8388608}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{1.5707963705062866} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999970197677612:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \sin 1.5707963705062866\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (if (<= (cos (* 6.28318530718 u2)) 0.9999970197677612)
  (* (sqrt u1) (sin (fma -6.28318530718 u2 1.5707963705062866)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin 1.5707963705062866))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9999970197677612f) {
		tmp = sqrtf(u1) * sinf(fmaf(-6.28318530718f, u2, 1.5707963705062866f));
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * sinf(1.5707963705062866f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9999970197677612))
		tmp = Float32(sqrt(u1) * sin(fma(Float32(-6.28318530718), u2, Float32(1.5707963705062866))));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(1.5707963705062866)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99999702

   1. Initial program 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{-314159265359}{50000000000}}, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      13. lower-PI.f32 99.2%

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \color{blue}{\pi}\right)\right) \]

   3. Applied rewrites 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right)} \]

   4. Evaluated real constant 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \color{blue}{1.5707963705062866}\right)\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 74.1%

         \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

   7. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

   if 0.99999702 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{-314159265359}{50000000000}}, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      13. lower-PI.f32 99.2%

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \color{blue}{\pi}\right)\right) \]

   3. Applied rewrites 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right)} \]

   4. Evaluated real constant 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \color{blue}{1.5707963705062866}\right)\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\frac{13176795}{8388608}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{1.5707963705062866} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.1% accurate, 1.1× speedup

Math

\[\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (* (sqrt u1) (sin (fma -6.28318530718 u2 1.5707963705062866))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * sinf(fmaf(-6.28318530718f, u2, 1.5707963705062866f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * sin(fma(Float32(-6.28318530718), u2, Float32(1.5707963705062866))))
end

Derivation

1. Initial program 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{-314159265359}{50000000000}}, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   9. mult-flip N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   13. lower-PI.f32 99.2%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \color{blue}{\pi}\right)\right) \]

3. Applied rewrites 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right)} \]

4. Evaluated real constant 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \color{blue}{1.5707963705062866}\right)\right) \]

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]
6. Step-by-step derivation

   1. lower-sqrt.f32 74.1%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

7. Applied rewrites 74.1%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]
8. Add Preprocessing

Alternative 5: 74.0% accurate, 1.1× speedup

Math

\[\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 74.0%

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

4. Applied rewrites 74.0%

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

Alternative 6: 63.5% accurate, 1.2× speedup

Math

\[\sqrt{u1} \cdot \sin 1.5707963705062866 \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (* (sqrt u1) (sin 1.5707963705062866)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * sinf(1.5707963705062866f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1) * sin(1.5707963705062866e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * sin(Float32(1.5707963705062866)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * sin(single(1.5707963705062866));
end

Derivation

1. Initial program 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\frac{-314159265359}{50000000000}}, u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   9. mult-flip N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(\frac{-314159265359}{50000000000}, u2, \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   13. lower-PI.f32 99.2%

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \color{blue}{\pi}\right)\right) \]

3. Applied rewrites 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right)} \]

4. Evaluated real constant 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, \color{blue}{1.5707963705062866}\right)\right) \]

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]
6. Step-by-step derivation

   1. lower-sqrt.f32 74.1%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

7. Applied rewrites 74.1%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

8. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\frac{13176795}{8388608}} \]
9. Step-by-step derivation

10. Applied rewrites 63.5%

    \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{1.5707963705062866} \]
11. Add Preprocessing

Alternative 7: 20.2% accurate, 1.3× speedup

Math

\[\cos \left(-6.28318530718 \cdot u2\right) \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. div-flip N/A

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

   7. *-commutative N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. neg-fabs N/A

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

   12. lower-fabs.f32 N/A

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

   13. lift--.f32 N/A

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

   14. sub-negate-rev N/A

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

   15. lower--.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

3. Applied rewrites 98.4%

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

4. Taylor expanded in u1 around inf

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

   1. lower-cos.f32 N/A

      \[\leadsto \cos \left(\frac{-314159265359}{50000000000} \cdot u2\right) \]

   2. lower-*.f32 20.2%

      \[\leadsto \cos \left(-6.28318530718 \cdot u2\right) \]

6. Applied rewrites 20.2%

   \[\leadsto \color{blue}{\cos \left(-6.28318530718 \cdot u2\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))