Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 9.3s

Alternatives: 8

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: 99.0% 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

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternative 2: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.05999999865889549)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2))))
   (* (cos (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.05999999865889549f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 * (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) :: tmp
    if ((6.28318530718e0 * u2) <= 0.05999999865889549e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.05999999865889549))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.05999999865889549))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0599999987

   1. Initial program 99.5%

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

   2. Taylor expanded in u2 around 0 99.4%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
   3. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-lft-identity 99.4%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{u1}{1 - u1}}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. associate-*r* 99.4%

         \[\leadsto 1 \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. distribute-rgt-out 99.4%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]

      5. unpow2 99.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

   4. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

   if 0.0599999987 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.2%

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

      1. clear-num 52.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

      2. associate-/r/ 52.1%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - u1} \cdot u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

   3. Applied egg-rr 98.0%

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

   4. Taylor expanded in u1 around 0 88.2%

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

      1. +-commutative 51.1%

         \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

   6. Simplified 88.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]

Alternative 3: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.15000000596046448)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2))))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.15000000596046448f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * u2)));
	} 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) <= 0.15000000596046448e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
    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(0.15000000596046448))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))));
	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(0.15000000596046448))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.150000006

   1. Initial program 99.5%

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

   2. Taylor expanded in u2 around 0 98.2%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
   3. Step-by-step derivation

      1. +-commutative 98.2%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-lft-identity 98.2%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{u1}{1 - u1}}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. associate-*r* 98.2%

         \[\leadsto 1 \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. distribute-rgt-out 98.3%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]

      5. unpow2 98.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

   4. Simplified 98.3%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

   if 0.150000006 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.8%

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

   2. Taylor expanded in u1 around 0 75.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 85.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u1 \cdot \left(u1 + 1\right)\\ \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.005200000014156103:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1 \cdot u1}{t_0}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u1 (+ u1 1.0))))
   (if (<= (/ u1 (- 1.0 u1)) 0.005200000014156103)
     (* (+ 1.0 (* -19.739208802181317 (* u2 u2))) (sqrt t_0))
     (pow (/ (- 1.0 (* u1 u1)) t_0) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 * (u1 + 1.0f);
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.005200000014156103f) {
		tmp = (1.0f + (-19.739208802181317f * (u2 * u2))) * sqrtf(t_0);
	} else {
		tmp = powf(((1.0f - (u1 * u1)) / t_0), -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) :: t_0
    real(4) :: tmp
    t_0 = u1 * (u1 + 1.0e0)
    if ((u1 / (1.0e0 - u1)) <= 0.005200000014156103e0) then
        tmp = (1.0e0 + ((-19.739208802181317e0) * (u2 * u2))) * sqrt(t_0)
    else
        tmp = ((1.0e0 - (u1 * u1)) / t_0) ** (-0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 * Float32(u1 + Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.005200000014156103))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))) * sqrt(t_0));
	else
		tmp = Float32(Float32(Float32(1.0) - Float32(u1 * u1)) / t_0) ^ Float32(-0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u1 * (u1 + single(1.0));
	tmp = single(0.0);
	if ((u1 / (single(1.0) - u1)) <= single(0.005200000014156103))
		tmp = (single(1.0) + (single(-19.739208802181317) * (u2 * u2))) * sqrt(t_0);
	else
		tmp = ((single(1.0) - (u1 * u1)) / t_0) ^ single(-0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 1 u1)) < 0.00520000001

   1. Initial program 99.2%

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

   2. Taylor expanded in u2 around 0 89.1%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
   3. Step-by-step derivation

      1. +-commutative 89.1%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-lft-identity 89.1%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{u1}{1 - u1}}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. associate-*r* 89.1%

         \[\leadsto 1 \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. distribute-rgt-out 89.1%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]

      5. unpow2 89.1%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

   4. Simplified 89.1%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]
   5. Step-by-step derivation

      1. clear-num 89.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

      2. associate-/r/ 89.1%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - u1} \cdot u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

   6. Applied egg-rr 89.1%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - u1} \cdot u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

   7. Taylor expanded in u1 around 0 87.9%

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1\right)} \cdot u1} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative 87.9%

         \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

   9. Simplified 87.9%

      \[\leadsto \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

   if 0.00520000001 < (/.f32 u1 (-.f32 1 u1))

   1. Initial program 99.2%

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

   2. Taylor expanded in u2 around 0 79.1%

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

      1. pow1/2 79.1%

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

      2. clear-num 79.0%

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

      3. inv-pow 79.0%

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

      4. pow-pow 79.1%

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

      5. div-sub 79.0%

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

      6. inv-pow 79.0%

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

      7. pow1 79.0%

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

      8. pow1 79.0%

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

      9. pow-div 79.0%

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

      10. metadata-eval 79.0%

          \[\leadsto {\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \]

      11. metadata-eval 79.0%

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

      12. metadata-eval 79.0%

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

   4. Applied egg-rr 79.0%

      \[\leadsto \color{blue}{{\left({u1}^{-1} - 1\right)}^{-0.5}} \]
   5. Step-by-step derivation

      1. unpow-1 79.0%

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

      2. sub-neg 79.0%

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

      3. metadata-eval 79.0%

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

   6. Simplified 79.0%

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

      1. metadata-eval 79.0%

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

      2. sub-neg 79.0%

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

      3. *-inverses 79.0%

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

      4. div-sub 79.1%

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

      5. div-inv 79.1%

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

      6. flip-- 79.0%

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

      7. frac-times 79.1%

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

      8. metadata-eval 79.1%

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

   8. Applied egg-rr 79.1%

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

      1. *-rgt-identity 79.1%

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

      2. *-commutative 79.1%

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

   10. Simplified 79.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.005200000014156103:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1 \cdot u1}{u1 \cdot \left(u1 + 1\right)}\right)}^{-0.5}\\ \end{array} \]

Alternative 5: 88.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* -19.739208802181317 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * (u2 * 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))) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
end

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in u2 around 0 89.4%

   \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
3. Step-by-step derivation

   1. +-commutative 89.4%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. *-lft-identity 89.4%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{u1}{1 - u1}}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   3. associate-*r* 89.4%

      \[\leadsto 1 \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. distribute-rgt-out 89.4%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]

   5. unpow2 89.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

4. Simplified 89.4%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

5. Final simplification 89.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

Alternative 6: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.008999999612569809:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.008999999612569809)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (+ 1.0 (* -19.739208802181317 (* u2 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.008999999612569809f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = (1.0f + (-19.739208802181317f * (u2 * 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 (u2 <= 0.008999999612569809e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = (1.0e0 + ((-19.739208802181317e0) * (u2 * u2))) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.008999999612569809))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.008999999612569809))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = (single(1.0) + (single(-19.739208802181317) * (u2 * u2))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00899999961

   1. Initial program 99.5%

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

   2. Taylor expanded in u2 around 0 93.8%

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

   if 0.00899999961 < u2

   1. Initial program 98.2%

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

   2. Taylor expanded in u2 around 0 52.1%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
   3. Step-by-step derivation

      1. +-commutative 52.1%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. *-lft-identity 52.1%

         \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{u1}{1 - u1}}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      3. associate-*r* 52.1%

         \[\leadsto 1 \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. distribute-rgt-out 52.1%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)} \]

      5. unpow2 52.1%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \]

   4. Simplified 52.1%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

   5. Taylor expanded in u1 around 0 47.8%

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.008999999612569809:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 7: 79.9% 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 99.2%

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

2. Taylor expanded in u2 around 0 80.6%

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

3. Final simplification 80.6%

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

Alternative 8: 63.1% 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 99.2%

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

2. Taylor expanded in u2 around 0 80.6%

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

3. Taylor expanded in u1 around 0 64.3%

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

4. Final simplification 64.3%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023207 
(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))))