Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 13.7s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. clear-num 99.0%

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

   2. sqrt-div 98.8%

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

   3. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

   1. div-sub 98.8%

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

   2. sub-neg 98.8%

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

   3. *-inverses 98.8%

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

   4. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. pow1/2 98.8%

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

   3. pow-flip 99.1%

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

   4. metadata-eval 99.1%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(1 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
9. Step-by-step derivation

   1. *-lft-identity 99.1%

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

   2. +-commutative 99.1%

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

   3. metadata-eval 99.1%

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

   4. distribute-neg-frac 99.1%

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

   5. unsub-neg 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 2: 94.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999960064888)
     (* t_0 (sqrt (* u1 (+ u1 1.0))))
     (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 <= 0.9999960064888f) {
		tmp = t_0 * sqrtf((u1 * (u1 + 1.0f)));
	} 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 <= 0.9999960064888e0) then
        tmp = t_0 * sqrt((u1 * (u1 + 1.0e0)))
    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(0.9999960064888))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	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(0.9999960064888))
		tmp = t_0 * sqrt((u1 * (u1 + single(1.0))));
	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)) < 0.999996006

   1. Initial program 98.5%

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

   3. Taylor expanded in u1 around 0 86.0%

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

      1. +-commutative 45.0%

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

   5. Simplified 86.0%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0 98.7%

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

4. Final simplification 94.2%

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

Alternative 3: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0038870000280439854:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0038870000280439854)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* 6.28318530718 u2)) (pow (/ 1.0 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0038870000280439854f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * powf((1.0f / u1), -0.5f);
	}
	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) <= 0.0038870000280439854e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((6.28318530718e0 * u2)) * ((1.0e0 / u1) ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0038870000280439854))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * (Float32(Float32(1.0) / u1) ^ Float32(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0038870000280439854))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((single(6.28318530718) * u2)) * ((single(1.0) / u1) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00388700003

   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 98.1%

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

   if 0.00388700003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.6%

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

      1. clear-num 98.4%

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

      2. sqrt-div 98.6%

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

      3. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. div-sub 98.5%

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

      2. sub-neg 98.5%

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

      3. *-inverses 98.5%

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

      4. metadata-eval 98.5%

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

   6. Simplified 98.5%

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

      1. *-un-lft-identity 98.5%

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

      2. pow1/2 98.5%

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

      3. pow-flip 98.5%

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

      4. metadata-eval 98.5%

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

   8. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\left(1 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 98.5%

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

      2. +-commutative 98.5%

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

      3. metadata-eval 98.5%

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

      4. distribute-neg-frac 98.5%

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

      5. unsub-neg 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in u1 around 0 75.4%

       \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0038870000280439854:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0038870000280439854:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0038870000280439854)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0038870000280439854f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} 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) <= 0.0038870000280439854e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    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(0.0038870000280439854))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	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(0.0038870000280439854))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	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) < 0.00388700003

   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 98.1%

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

   if 0.00388700003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.6%

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

   3. Taylor expanded in u1 around 0 75.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0038870000280439854:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.1%

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

3. Final simplification 99.1%

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

Alternative 6: 72.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (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((u1 * (u1 + 1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 80.3%

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

4. Taylor expanded in u1 around 0 70.7%

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

   1. +-commutative 70.7%

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

6. Simplified 70.7%

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

7. Final simplification 70.7%

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

Alternative 7: 72.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 + (u1 * 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 + (u1 * u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0 80.3%

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

4. Taylor expanded in u1 around 0 70.7%

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

   1. distribute-rgt-in 70.8%

      \[\leadsto \sqrt{\color{blue}{1 \cdot u1 + u1 \cdot u1}} \]

   2. *-lft-identity 70.8%

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

   3. unpow2 70.8%

      \[\leadsto \sqrt{u1 + \color{blue}{{u1}^{2}}} \]

6. Simplified 70.8%

   \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 70.8%

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

8. Applied egg-rr 70.8%

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

9. Final simplification 70.8%

   \[\leadsto \sqrt{u1 + u1 \cdot u1} \]
10. Add Preprocessing

Alternative 8: 80.5% 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 99.1%

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

3. Taylor expanded in u2 around 0 80.3%

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

4. Final simplification 80.3%

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

Alternative 9: 63.8% 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 99.1%

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

3. Taylor expanded in u2 around 0 80.3%

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

4. Taylor expanded in u1 around 0 62.5%

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

5. Final simplification 62.5%

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

Alternative 10: 20.1% 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 99.1%

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

3. Taylor expanded in u2 around 0 80.3%

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

4. Taylor expanded in u1 around 0 70.7%

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

   1. +-commutative 70.7%

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

6. Simplified 70.7%

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

7. Taylor expanded in u1 around inf 20.8%

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

   1. associate-*r/ 20.8%

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

   2. metadata-eval 20.8%

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

9. Simplified 20.8%

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

10. Final simplification 20.8%

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

Alternative 11: 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 99.1%

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

3. Taylor expanded in u2 around 0 80.3%

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

4. Taylor expanded in u1 around 0 70.7%

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

   1. distribute-rgt-in 70.8%

      \[\leadsto \sqrt{\color{blue}{1 \cdot u1 + u1 \cdot u1}} \]

   2. *-lft-identity 70.8%

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

   3. unpow2 70.8%

      \[\leadsto \sqrt{u1 + \color{blue}{{u1}^{2}}} \]

6. Simplified 70.8%

   \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \]

7. Taylor expanded in u1 around -inf 4.3%

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

   1. neg-mul-1 4.3%

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

9. Simplified 4.3%

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

10. Final simplification 4.3%

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

Reproduce

herbie shell --seed 2024059 
(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))))