Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.1%

Time: 6.8s

Alternatives: 7

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

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

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)
    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

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

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)
    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: 99.1% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. neg-sub0 N/A

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

   9. lift--.f32 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. associate--r+ N/A

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

   13. neg-sub0 N/A

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

   14. remove-double-neg N/A

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

   15. lower--.f32 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-*.f32 N/A

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

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

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

   5. metadata-eval N/A

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

   6. lift-*.f32 N/A

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

   7. lift-cos.f32 98.6

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

   8. remove-double-neg N/A

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

   9. lift-cos.f32 N/A

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

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

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

   11. sin-neg-rev N/A

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

   12. sin-neg-rev N/A

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

   13. distribute-neg-in N/A

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

   14. lift-*.f32 N/A

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

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

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

   16. metadata-eval N/A

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

   17. lift-*.f32 N/A

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

   18. sub-neg N/A

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

   19. lower-sin.f32 N/A

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

   20. lower-neg.f32 N/A

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

   21. lower--.f32 N/A

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

6. Applied rewrites 98.7%

   \[\leadsto \sqrt{\frac{-1}{u1 - 1} \cdot u1} \cdot \color{blue}{\sin \left(-\left(u2 \cdot 6.28318530718 - 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. un-div-inv N/A

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

   7. lift--.f32 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-in N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. sub-neg N/A

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

   14. lift--.f32 N/A

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

   15. lower-/.f32 98.9

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

   \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot 0.5 - 6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
10. Add Preprocessing

Alternative 2: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999399781227112:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999399781227112)
     (* (sqrt u1) t_0)
     (sqrt (/ u1 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999399781227112f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999399781227112e0) then
        tmp = sqrt(u1) * t_0
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999399781227112))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999399781227112))
		tmp = sqrt(u1) * t_0;
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999939978

   1. Initial program 97.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 74.1

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

   5. Applied rewrites 74.1%

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

   if 0.999939978 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

      2. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

      3. lower--.f32 96.0

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999399781227112:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.001500000013038516)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (sqrt (* (- u1 -1.0) u1)) (cos (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.001500000013038516f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(((u1 - -1.0f) * u1)) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.001500000013038516e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(((u1 - (-1.0e0)) * u1)) * cos((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.001500000013038516))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(sqrt(Float32(Float32(u1 - Float32(-1.0)) * u1)) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.001500000013038516))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(((u1 - single(-1.0)) * u1)) * cos((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00150000001

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

      2. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

      3. lower--.f32 98.6

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   if 0.00150000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-+.f32 85.0

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

   5. Applied rewrites 85.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((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 = cos((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Final simplification 98.7%

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

Alternative 5: 80.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   2. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   3. lower--.f32 78.0

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \]

5. Applied rewrites 78.0%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
6. Add Preprocessing

Alternative 6: 71.8% accurate, 7.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(((u1 - (-1.0e0)) * u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   2. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   3. lower--.f32 78.0

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \]

5. Applied rewrites 78.0%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around inf

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

8. Applied rewrites 77.9%

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

9. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 71.1%

    \[\leadsto \sqrt{\left(1 + u1\right) \cdot u1} \]

12. Final simplification 71.1%

    \[\leadsto \sqrt{\left(u1 - -1\right) \cdot u1} \]
13. Add Preprocessing

Alternative 7: 63.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   2. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   3. lower--.f32 78.0

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \]

5. Applied rewrites 78.0%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
7. Step-by-step derivation

8. Applied rewrites 63.2%

   \[\leadsto \sqrt{u1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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))))