Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 35.4s

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

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

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)
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))) * 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

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

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)
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))) * 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: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. distribute-neg-frac N/A

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

   5. div-flip N/A

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

   6. remove-double-neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 98.3%

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

3. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\sqrt{\frac{u1}{0.5 - \left(u1 - 0.5\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(0.5) - (u1 - single(0.5))))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.3%

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

   1. +-rgt-identity N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. sub-flip N/A

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

   7. lift--.f32 N/A

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

   8. metadata-eval N/A

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

   9. associate-+r- N/A

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

   10. lower--.f32 N/A

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

   11. lower-+.f32 98.0%

       \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(\left(1 - u1\right) + 3\right)} - 3}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied rewrites 98.0%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--l+ N/A

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

   4. metadata-eval N/A

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

   5. +-rgt-identity 98.3%

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

   6. lift--.f32 N/A

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

   7. sub-flip N/A

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

   8. add-flip N/A

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

   9. sub-negate-rev N/A

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

   10. remove-double-neg N/A

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

   11. *-lft-identity N/A

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

   12. *-inverses N/A

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

   13. associate-*l/ N/A

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

   14. *-inverses N/A

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

   15. sub-div N/A

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

   16. sub-to-fraction N/A

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

   17. *-inverses N/A

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

   18. metadata-eval N/A

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

   19. associate--r+ N/A

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

   20. sub-negate N/A

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

   21. lower--.f32 N/A

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

   22. lower--.f32 98.4%

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

5. Applied rewrites 98.4%

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

Alternative 3: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.00039999998989515007:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\left|u1 - 1\right|}}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\left|u1\right|}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (if (<= u2 0.00039999998989515007)
  (/
   1.0
   (/
    (sqrt (fabs (- u1 1.0)))
    (* 6.28318530718 (* u2 (sqrt (fabs u1))))))
  (* (sqrt (* u1 (+ 1.0 u1))) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.00039999998989515007f) {
		tmp = 1.0f / (sqrtf(fabsf((u1 - 1.0f))) / (6.28318530718f * (u2 * sqrtf(fabsf(u1)))));
	} else {
		tmp = sqrtf((u1 * (1.0f + u1))) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (u2 <= 0.00039999998989515007e0) then
        tmp = 1.0e0 / (sqrt(abs((u1 - 1.0e0))) / (6.28318530718e0 * (u2 * sqrt(abs(u1)))))
    else
        tmp = sqrt((u1 * (1.0e0 + u1))) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.00039999998989515007))
		tmp = Float32(Float32(1.0) / Float32(sqrt(abs(Float32(u1 - Float32(1.0)))) / Float32(Float32(6.28318530718) * Float32(u2 * sqrt(abs(u1))))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + u1))) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.00039999998989515007))
		tmp = single(1.0) / (sqrt(abs((u1 - single(1.0)))) / (single(6.28318530718) * (u2 * sqrt(abs(u1)))));
	else
		tmp = sqrt((u1 * (single(1.0) + u1))) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 3.9999999e-4

   1. Initial program 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. div-flip N/A

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

      7. *-commutative N/A

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

      8. lower-/.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. neg-fabs N/A

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

      12. lower-fabs.f32 N/A

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

      13. lift--.f32 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-fabs.f32 81.6%

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

   6. Applied rewrites 81.6%

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

   if 3.9999999e-4 < u2

   1. Initial program 98.3%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 86.1%

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

   4. Applied rewrites 86.1%

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

Alternative 4: 89.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.0010000000474974513:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\left|u1 - 1\right|}}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\left|u1\right|}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (if (<= u2 0.0010000000474974513)
  (/
   1.0
   (/
    (sqrt (fabs (- u1 1.0)))
    (* 6.28318530718 (* u2 (sqrt (fabs u1))))))
  (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0010000000474974513f) {
		tmp = 1.0f / (sqrtf(fabsf((u1 - 1.0f))) / (6.28318530718f * (u2 * sqrtf(fabsf(u1)))));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (u2 <= 0.0010000000474974513e0) then
        tmp = 1.0e0 / (sqrt(abs((u1 - 1.0e0))) / (6.28318530718e0 * (u2 * sqrt(abs(u1)))))
    else
        tmp = sqrt(u1) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(1.0) / Float32(sqrt(abs(Float32(u1 - Float32(1.0)))) / Float32(Float32(6.28318530718) * Float32(u2 * sqrt(abs(u1))))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.0010000000474974513))
		tmp = single(1.0) / (sqrt(abs((u1 - single(1.0)))) / (single(6.28318530718) * (u2 * sqrt(abs(u1)))));
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00100000005

   1. Initial program 98.3%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. associate-*l/ N/A

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

      6. div-flip N/A

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

      7. *-commutative N/A

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

      8. lower-/.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. neg-fabs N/A

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

      12. lower-fabs.f32 N/A

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

      13. lift--.f32 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

   3. Applied rewrites 98.0%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-fabs.f32 81.6%

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

   6. Applied rewrites 81.6%

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

   if 0.00100000005 < u2

   1. Initial program 98.3%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 74.0%

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

   4. Applied rewrites 74.0%

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

Alternative 5: 81.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 = 1.0e0 / (sqrt(abs((u1 - 1.0e0))) / (6.28318530718e0 * (u2 * sqrt(abs(u1)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. div-flip N/A

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

   7. *-commutative N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. neg-fabs N/A

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

   12. lower-fabs.f32 N/A

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

   13. lift--.f32 N/A

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

   14. sub-negate-rev N/A

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

   15. lower--.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

3. Applied rewrites 98.0%

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

4. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-fabs.f32 81.6%

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

6. Applied rewrites 81.6%

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

Alternative 6: 74.6% accurate, 2.1× speedup

Math

\[\frac{1}{\frac{1 + -0.5 \cdot u1}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\left|u1\right|}\right)}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (/
 1.0
 (/ (+ 1.0 (* -0.5 u1)) (* 6.28318530718 (* u2 (sqrt (fabs u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f / ((1.0f + (-0.5f * u1)) / (6.28318530718f * (u2 * sqrtf(fabsf(u1)))));
}

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 = 1.0e0 / ((1.0e0 + ((-0.5e0) * u1)) / (6.28318530718e0 * (u2 * sqrt(abs(u1)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + Float32(Float32(-0.5) * u1)) / Float32(Float32(6.28318530718) * Float32(u2 * sqrt(abs(u1))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) / ((single(1.0) + (single(-0.5) * u1)) / (single(6.28318530718) * (u2 * sqrt(abs(u1)))));
end

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. div-flip N/A

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

   7. *-commutative N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. neg-fabs N/A

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

   12. lower-fabs.f32 N/A

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

   13. lift--.f32 N/A

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

   14. sub-negate-rev N/A

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

   15. lower--.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

3. Applied rewrites 98.0%

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

4. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-fabs.f32 81.6%

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

6. Applied rewrites 81.6%

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

7. Taylor expanded in u1 around 0

   \[\leadsto \frac{1}{\frac{\color{blue}{1 + \frac{-1}{2} \cdot u1}}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\left|u1\right|}\right)}} \]
8. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 74.6%

      \[\leadsto \frac{1}{\frac{1 + -0.5 \cdot \color{blue}{u1}}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\left|u1\right|}\right)}} \]

9. Applied rewrites 74.6%

   \[\leadsto \frac{1}{\frac{\color{blue}{1 + -0.5 \cdot u1}}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\left|u1\right|}\right)}} \]
10. Add Preprocessing

Alternative 7: 64.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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 = 1.0e0 / (sqrt(abs((-1.0e0))) / (6.28318530718e0 * (u2 * sqrt(abs(u1)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. sqrt-div N/A

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

   5. associate-*l/ N/A

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

   6. div-flip N/A

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

   7. *-commutative N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

   11. neg-fabs N/A

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

   12. lower-fabs.f32 N/A

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

   13. lift--.f32 N/A

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

   14. sub-negate-rev N/A

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

   15. lower--.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

3. Applied rewrites 98.0%

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

4. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-fabs.f32 81.6%

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

6. Applied rewrites 81.6%

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

7. Taylor expanded in u1 around 0

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

9. Applied rewrites 64.7%

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

Reproduce

herbie shell --seed 2025322 
(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))))