Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 13.5s

Alternatives: 10

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. sqrt-div 98.5%

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

   3. associate-*r/ 98.4%

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

4. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-/l* 98.3%

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

6. Simplified 98.3%

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

   1. associate-*r/ 98.4%

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

   2. associate-*l/ 98.5%

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

   3. sqrt-div 98.8%

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

   4. add-sqr-sqrt 92.5%

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

   5. sqrt-unprod 93.7%

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

   6. swap-sqr 93.7%

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

   7. add-sqr-sqrt 93.7%

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

   8. pow2 93.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}} \]

8. Applied egg-rr 93.7%

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

   1. *-commutative 93.7%

      \[\leadsto \sqrt{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{u1}{1 - u1}}} \]

   2. *-lft-identity 93.7%

      \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{\color{blue}{1 \cdot u1}}{1 - u1}} \]

   3. associate-*l/ 93.6%

      \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\left(\frac{1}{1 - u1} \cdot u1\right)}} \]

   4. associate-/r/ 93.7%

      \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   5. associate-*r/ 93.7%

      \[\leadsto \sqrt{\color{blue}{\frac{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot 1}{\frac{1 - u1}{u1}}}} \]

   6. *-rgt-identity 93.7%

      \[\leadsto \sqrt{\frac{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}}{\frac{1 - u1}{u1}}} \]

   7. *-commutative 93.7%

      \[\leadsto \sqrt{\frac{{\cos \color{blue}{\left(u2 \cdot 6.28318530718\right)}}^{2}}{\frac{1 - u1}{u1}}} \]

   8. div-sub 93.8%

      \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

   9. sub-neg 93.8%

      \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} + \left(-\frac{u1}{u1}\right)}}} \]

   10. *-inverses 93.8%

       \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \left(-\color{blue}{1}\right)}} \]

   11. metadata-eval 93.8%

       \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \color{blue}{-1}}} \]

10. Simplified 93.8%

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

    1. div-inv 93.8%

       \[\leadsto \sqrt{\color{blue}{{\cos \left(u2 \cdot 6.28318530718\right)}^{2} \cdot \frac{1}{\frac{1}{u1} + -1}}} \]

    2. sqrt-prod 93.7%

       \[\leadsto \color{blue}{\sqrt{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}} \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}} \]

    3. *-commutative 93.7%

       \[\leadsto \sqrt{{\cos \color{blue}{\left(6.28318530718 \cdot u2\right)}}^{2}} \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}} \]

    4. sqrt-pow1 98.8%

       \[\leadsto \color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}} \]

    5. metadata-eval 98.8%

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

    6. pow1 98.8%

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

    7. metadata-eval 98.8%

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

    8. sub-neg 98.8%

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

    9. *-inverses 98.8%

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

    10. div-sub 98.7%

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

    11. *-commutative 98.7%

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

12. Applied egg-rr 98.9%

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

13. Final simplification 98.9%

    \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
14. Add Preprocessing

Alternative 2: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999960064888:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999960064888)
     (* t_0 (sqrt (* u1 (+ 1.0 u1))))
     (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999960064888f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + u1)));
	} else {
		tmp = sqrtf((1.0f / ((1.0f / u1) + -1.0f)));
	}
	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 <= 0.9999960064888e0) then
        tmp = t_0 * sqrt((u1 * (1.0e0 + u1)))
    else
        tmp = sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0))))
    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(0.9999960064888))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	else
		tmp = sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	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(0.9999960064888))
		tmp = t_0 * sqrt((u1 * (single(1.0) + u1)));
	else
		tmp = sqrt((single(1.0) / ((single(1.0) / u1) + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999996006

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 84.9%

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

      1. +-commutative 40.2%

         \[\leadsto e^{\log \left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right) \cdot 0.5} \]

   5. Simplified 84.9%

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

   if 0.999996006 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. sqrt-div 99.0%

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

      3. associate-*r/ 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-/l* 98.7%

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

   6. Simplified 98.7%

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

      1. associate-*r/ 99.0%

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

      2. associate-*l/ 99.0%

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

      3. sqrt-div 99.3%

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

      4. add-sqr-sqrt 98.1%

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

      5. sqrt-unprod 99.3%

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

      6. swap-sqr 99.3%

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

      7. add-sqr-sqrt 99.4%

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

      8. pow2 99.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}} \]

   8. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

         \[\leadsto \sqrt{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{u1}{1 - u1}}} \]

      2. *-lft-identity 99.4%

         \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{\color{blue}{1 \cdot u1}}{1 - u1}} \]

      3. associate-*l/ 99.2%

         \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\left(\frac{1}{1 - u1} \cdot u1\right)}} \]

      4. associate-/r/ 99.2%

         \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      5. associate-*r/ 99.2%

         \[\leadsto \sqrt{\color{blue}{\frac{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot 1}{\frac{1 - u1}{u1}}}} \]

      6. *-rgt-identity 99.2%

         \[\leadsto \sqrt{\frac{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}}{\frac{1 - u1}{u1}}} \]

      7. *-commutative 99.2%

         \[\leadsto \sqrt{\frac{{\cos \color{blue}{\left(u2 \cdot 6.28318530718\right)}}^{2}}{\frac{1 - u1}{u1}}} \]

      8. div-sub 99.4%

         \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

      9. sub-neg 99.4%

         \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} + \left(-\frac{u1}{u1}\right)}}} \]

      10. *-inverses 99.4%

          \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \left(-\color{blue}{1}\right)}} \]

      11. metadata-eval 99.4%

          \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \color{blue}{-1}}} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + -1}}} \]

   11. Taylor expanded in u2 around 0 98.3%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999960064888:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005200000014156103:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005200000014156103)
   (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0)))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.005200000014156103f) {
		tmp = sqrtf((1.0f / ((1.0f / u1) + -1.0f)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(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) :: tmp
    if ((6.28318530718e0 * u2) <= 0.005200000014156103e0) then
        tmp = sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0))))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.005200000014156103))
		tmp = sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.005200000014156103))
		tmp = sqrt((single(1.0) / ((single(1.0) / u1) + single(-1.0))));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00520000001

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. sqrt-div 98.9%

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

      3. associate-*r/ 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-/l* 98.7%

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

   6. Simplified 98.7%

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

      1. associate-*r/ 99.0%

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

      2. associate-*l/ 98.9%

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

      3. sqrt-div 99.3%

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

      4. add-sqr-sqrt 98.2%

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

      5. sqrt-unprod 99.3%

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

      6. swap-sqr 99.3%

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

      7. add-sqr-sqrt 99.4%

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

      8. pow2 99.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}} \]

   8. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

         \[\leadsto \sqrt{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{u1}{1 - u1}}} \]

      2. *-lft-identity 99.4%

         \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{\color{blue}{1 \cdot u1}}{1 - u1}} \]

      3. associate-*l/ 99.2%

         \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\left(\frac{1}{1 - u1} \cdot u1\right)}} \]

      4. associate-/r/ 99.3%

         \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      5. associate-*r/ 99.3%

         \[\leadsto \sqrt{\color{blue}{\frac{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot 1}{\frac{1 - u1}{u1}}}} \]

      6. *-rgt-identity 99.3%

         \[\leadsto \sqrt{\frac{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}}{\frac{1 - u1}{u1}}} \]

      7. *-commutative 99.3%

         \[\leadsto \sqrt{\frac{{\cos \color{blue}{\left(u2 \cdot 6.28318530718\right)}}^{2}}{\frac{1 - u1}{u1}}} \]

      8. div-sub 99.4%

         \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

      9. sub-neg 99.4%

         \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} + \left(-\frac{u1}{u1}\right)}}} \]

      10. *-inverses 99.4%

          \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \left(-\color{blue}{1}\right)}} \]

      11. metadata-eval 99.4%

          \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \color{blue}{-1}}} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + -1}}} \]

   11. Taylor expanded in u2 around 0 98.0%

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

   if 0.00520000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 72.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005200000014156103:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
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 98.8%

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

3. Final simplification 98.8%

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

Alternative 5: 80.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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((1.0e0 / ((1.0e0 / u1) + (-1.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. *-commutative 98.8%

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

   2. sqrt-div 98.5%

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

   3. associate-*r/ 98.4%

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

4. Applied egg-rr 98.4%

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

   1. *-commutative 98.4%

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

   2. associate-/l* 98.3%

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

6. Simplified 98.3%

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

   1. associate-*r/ 98.4%

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

   2. associate-*l/ 98.5%

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

   3. sqrt-div 98.8%

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

   4. add-sqr-sqrt 92.5%

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

   5. sqrt-unprod 93.7%

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

   6. swap-sqr 93.7%

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

   7. add-sqr-sqrt 93.7%

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

   8. pow2 93.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}} \]

8. Applied egg-rr 93.7%

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

   1. *-commutative 93.7%

      \[\leadsto \sqrt{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{u1}{1 - u1}}} \]

   2. *-lft-identity 93.7%

      \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \frac{\color{blue}{1 \cdot u1}}{1 - u1}} \]

   3. associate-*l/ 93.6%

      \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\left(\frac{1}{1 - u1} \cdot u1\right)}} \]

   4. associate-/r/ 93.7%

      \[\leadsto \sqrt{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   5. associate-*r/ 93.7%

      \[\leadsto \sqrt{\color{blue}{\frac{{\cos \left(6.28318530718 \cdot u2\right)}^{2} \cdot 1}{\frac{1 - u1}{u1}}}} \]

   6. *-rgt-identity 93.7%

      \[\leadsto \sqrt{\frac{\color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}}{\frac{1 - u1}{u1}}} \]

   7. *-commutative 93.7%

      \[\leadsto \sqrt{\frac{{\cos \color{blue}{\left(u2 \cdot 6.28318530718\right)}}^{2}}{\frac{1 - u1}{u1}}} \]

   8. div-sub 93.8%

      \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

   9. sub-neg 93.8%

      \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\color{blue}{\frac{1}{u1} + \left(-\frac{u1}{u1}\right)}}} \]

   10. *-inverses 93.8%

       \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \left(-\color{blue}{1}\right)}} \]

   11. metadata-eval 93.8%

       \[\leadsto \sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + \color{blue}{-1}}} \]

10. Simplified 93.8%

    \[\leadsto \color{blue}{\sqrt{\frac{{\cos \left(u2 \cdot 6.28318530718\right)}^{2}}{\frac{1}{u1} + -1}}} \]

11. Taylor expanded in u2 around 0 78.1%

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

12. Final simplification 78.1%

    \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} + -1}} \]
13. Add Preprocessing

Alternative 6: 71.7% accurate, 2.0× speedup

Math

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

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 98.8%

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

3. Taylor expanded in u2 around 0 78.1%

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

4. Taylor expanded in u1 around 0 70.5%

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

   1. +-commutative 69.3%

      \[\leadsto e^{\log \left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right) \cdot 0.5} \]

6. Simplified 70.5%

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

7. Final simplification 70.5%

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

Alternative 7: 80.2% accurate, 2.0× 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 98.8%

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

3. Taylor expanded in u2 around 0 78.1%

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

4. Final simplification 78.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 2.1× 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 98.8%

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

3. Taylor expanded in u2 around 0 78.1%

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

4. Taylor expanded in u1 around 0 61.8%

   \[\leadsto \color{blue}{\sqrt{u1}} \]

5. Final simplification 61.8%

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

Alternative 9: 20.2% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \left(1 + \frac{0.5}{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* u1 (+ 1.0 (/ 0.5 u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * (1.0f + (0.5f / 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 = u1 * (1.0e0 + (0.5e0 / u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in u2 around 0 78.1%

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

4. Taylor expanded in u1 around 0 70.5%

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

   1. +-commutative 69.3%

      \[\leadsto e^{\log \left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right) \cdot 0.5} \]

6. Simplified 70.5%

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

7. Taylor expanded in u1 around inf 20.2%

   \[\leadsto \color{blue}{u1 \cdot \left(1 + 0.5 \cdot \frac{1}{u1}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 20.2%

      \[\leadsto u1 \cdot \left(1 + \color{blue}{\frac{0.5 \cdot 1}{u1}}\right) \]

   2. metadata-eval 20.2%

      \[\leadsto u1 \cdot \left(1 + \frac{\color{blue}{0.5}}{u1}\right) \]

9. Simplified 20.2%

   \[\leadsto \color{blue}{u1 \cdot \left(1 + \frac{0.5}{u1}\right)} \]

10. Final simplification 20.2%

    \[\leadsto u1 \cdot \left(1 + \frac{0.5}{u1}\right) \]
11. Add Preprocessing

Alternative 10: 4.4% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in u2 around 0 78.1%

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

   1. pow1/2 78.1%

      \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}} \]

   2. pow-to-exp 76.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{u1}{1 - u1}\right) \cdot 0.5}} \]

5. Applied egg-rr 76.6%

   \[\leadsto \color{blue}{e^{\log \left(\frac{u1}{1 - u1}\right) \cdot 0.5}} \]

6. Taylor expanded in u1 around 0 69.3%

   \[\leadsto e^{\log \color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)} \cdot 0.5} \]
7. Step-by-step derivation

   1. +-commutative 69.3%

      \[\leadsto e^{\log \left(u1 \cdot \color{blue}{\left(u1 + 1\right)}\right) \cdot 0.5} \]

8. Simplified 69.3%

   \[\leadsto e^{\log \color{blue}{\left(u1 \cdot \left(u1 + 1\right)\right)} \cdot 0.5} \]

9. Taylor expanded in u1 around -inf 4.5%

   \[\leadsto \color{blue}{-1 \cdot u1} \]
10. Step-by-step derivation

    1. neg-mul-1 4.5%

       \[\leadsto \color{blue}{-u1} \]

11. Simplified 4.5%

    \[\leadsto \color{blue}{-u1} \]

12. Final simplification 4.5%

    \[\leadsto -u1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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))))