Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 14.8s

Alternatives: 13

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (sqrt (* (* u2 u2) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf(sqrtf(((u2 * u2) * 39.47841760436263f)));
}

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(sqrt(((u2 * u2) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(sqrt(Float32(Float32(u2 * u2) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos(sqrt(((u2 * u2) * single(39.47841760436263))));
end

Derivation

1. Initial program 99.1%

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

   1. add-sqr-sqrt 98.9%

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

   2. pow1/2 98.9%

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

   3. pow1/2 98.9%

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

   4. pow-prod-down 99.1%

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

   5. swap-sqr 99.0%

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

   6. metadata-eval 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 99.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

   2. unpow2 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{39.47841760436263 \cdot \color{blue}{{u2}^{2}}}\right) \]

   3. *-commutative 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}\right) \]

   4. unpow2 99.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right)} \cdot 39.47841760436263}\right) \]

5. Simplified 99.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)} \]

6. Final simplification 99.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

Alternative 2: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9884999990463257:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9884999990463257)
     (* t_0 (sqrt u1))
     (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9884999990463257f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	}
	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((u2 * 6.28318530718e0))
    if (t_0 <= 0.9884999990463257e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9884999990463257))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.9884999990463257))
		tmp = t_0 * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.988499999

   1. Initial program 98.2%

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

   2. Taylor expanded in u1 around 0 76.8%

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

   if 0.988499999 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   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 98.4%

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

      1. *-rgt-identity 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. distribute-lft-out 98.4%

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

      5. unpow2 98.4%

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

   4. Simplified 98.4%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9884999990463257:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \]

Alternative 3: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999334812164307:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* u2 6.28318530718)) 0.9999334812164307)
   (* (sqrt u1) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (sqrt (* (/ u1 (- 1.0 (* u1 u1))) (+ u1 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((u2 * 6.28318530718f)) <= 0.9999334812164307f) {
		tmp = sqrtf(u1) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = sqrtf(((u1 / (1.0f - (u1 * 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 (cos((u2 * 6.28318530718e0)) <= 0.9999334812164307e0) then
        tmp = sqrt(u1) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = sqrt(((u1 / (1.0e0 - (u1 * u1))) * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(u2 * Float32(6.28318530718))) <= Float32(0.9999334812164307))
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - Float32(u1 * u1))) * Float32(u1 + Float32(1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (cos((u2 * single(6.28318530718))) <= single(0.9999334812164307))
		tmp = sqrt(u1) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = sqrt(((u1 / (single(1.0) - (u1 * u1))) * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999933481

   1. Initial program 98.6%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in u2 around 0 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in u1 around 0 53.9%

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

   if 0.999933481 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.3%

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

   2. Taylor expanded in u2 around 0 97.1%

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

      1. flip-- 97.2%

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

      2. associate-/r/ 97.2%

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

      3. metadata-eval 97.2%

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

      4. +-commutative 97.2%

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

   4. Applied egg-rr 97.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999334812164307:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)}\\ \end{array} \]

Alternative 4: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999334812164307:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* u2 6.28318530718)) 0.9999334812164307)
   (* (sqrt u1) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((u2 * 6.28318530718f)) <= 0.9999334812164307f) {
		tmp = sqrtf(u1) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} 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) :: tmp
    if (cos((u2 * 6.28318530718e0)) <= 0.9999334812164307e0) then
        tmp = sqrt(u1) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(u2 * Float32(6.28318530718))) <= Float32(0.9999334812164307))
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (cos((u2 * single(6.28318530718))) <= single(0.9999334812164307))
		tmp = sqrt(u1) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	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 314159265359/50000000000 u2)) < 0.999933481

   1. Initial program 98.6%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in u2 around 0 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in u1 around 0 53.9%

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

   if 0.999933481 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.3%

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

   2. Taylor expanded in u2 around 0 97.1%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999334812164307:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]

Alternative 5: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.15199999511241913:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.15199999511241913)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* u2 (* u2 -19.739208802181317))))
   (* (cos (* u2 6.28318530718)) (pow (/ 1.0 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.15199999511241913f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.15199999511241913e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
    else
        tmp = cos((u2 * 6.28318530718e0)) * ((1.0e0 / u1) ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.15199999511241913))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * (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 ((u2 * single(6.28318530718)) <= single(0.15199999511241913))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
	else
		tmp = cos((u2 * single(6.28318530718))) * ((single(1.0) / u1) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.151999995

   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 98.4%

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

      1. *-rgt-identity 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. distribute-lft-out 98.4%

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

      5. unpow2 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in u2 around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. unpow2 98.4%

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

      3. associate-*r* 98.4%

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

   7. Simplified 98.4%

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

   if 0.151999995 < (*.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. add-sqr-sqrt 97.1%

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

      2. pow1/2 97.1%

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

      3. pow1/2 97.1%

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

      4. pow-prod-down 98.2%

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

      5. swap-sqr 98.0%

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

      6. metadata-eval 98.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

   3. Applied egg-rr 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
   4. Step-by-step derivation

      1. unpow1/2 98.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

      2. unpow2 98.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{39.47841760436263 \cdot \color{blue}{{u2}^{2}}}\right) \]

      3. *-commutative 98.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}\right) \]

      4. unpow2 98.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right)} \cdot 39.47841760436263}\right) \]

   5. Simplified 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)} \]
   6. Step-by-step derivation

      1. pow1/2 98.3%

         \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      2. clear-num 98.2%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{0.5} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      3. inv-pow 98.2%

         \[\leadsto {\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{0.5} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      4. pow-pow 98.2%

         \[\leadsto \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot 0.5\right)}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      5. div-sub 98.2%

         \[\leadsto {\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\left(-1 \cdot 0.5\right)} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      6. pow1 98.2%

         \[\leadsto {\left(\frac{1}{u1} - \frac{\color{blue}{{u1}^{1}}}{u1}\right)}^{\left(-1 \cdot 0.5\right)} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      7. pow1 98.2%

         \[\leadsto {\left(\frac{1}{u1} - \frac{{u1}^{1}}{\color{blue}{{u1}^{1}}}\right)}^{\left(-1 \cdot 0.5\right)} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      8. pow-div 98.2%

         \[\leadsto {\left(\frac{1}{u1} - \color{blue}{{u1}^{\left(1 - 1\right)}}\right)}^{\left(-1 \cdot 0.5\right)} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      10. metadata-eval 98.2%

          \[\leadsto {\left(\frac{1}{u1} - \color{blue}{1}\right)}^{\left(-1 \cdot 0.5\right)} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

      11. metadata-eval 98.2%

          \[\leadsto {\left(\frac{1}{u1} - 1\right)}^{\color{blue}{-0.5}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

   7. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} - 1\right)}^{-0.5}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

   8. Taylor expanded in u2 around 0 98.0%

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

      1. *-commutative 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in u1 around 0 76.9%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.15199999511241913:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

2. Final simplification 99.1%

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

Alternative 7: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0006500000017695129)
   (sqrt (* (/ u1 (- 1.0 (* u1 u1))) (+ u1 1.0)))
   (* (+ 1.0 (* (* u2 u2) -19.739208802181317)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0006500000017695129f) {
		tmp = sqrtf(((u1 / (1.0f - (u1 * u1))) * (u1 + 1.0f)));
	} else {
		tmp = (1.0f + ((u2 * u2) * -19.739208802181317f)) * 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 ((u2 * 6.28318530718e0) <= 0.0006500000017695129e0) then
        tmp = sqrt(((u1 / (1.0e0 - (u1 * u1))) * (u1 + 1.0e0)))
    else
        tmp = (1.0e0 + ((u2 * u2) * (-19.739208802181317e0))) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0006500000017695129))
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - Float32(u1 * u1))) * Float32(u1 + Float32(1.0))));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))) * 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 ((u2 * single(6.28318530718)) <= single(0.0006500000017695129))
		tmp = sqrt(((u1 / (single(1.0) - (u1 * u1))) * (u1 + single(1.0))));
	else
		tmp = (single(1.0) + ((u2 * u2) * single(-19.739208802181317))) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 6.50000002e-4

   1. Initial program 99.4%

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

   2. Taylor expanded in u2 around 0 99.3%

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

      1. flip-- 99.3%

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

      2. associate-/r/ 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

   4. Applied egg-rr 99.3%

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

   if 6.50000002e-4 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   2. 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. associate-/r/ 98.4%

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

   3. Applied egg-rr 98.4%

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

   4. Taylor expanded in u2 around 0 72.2%

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

      1. *-commutative 72.2%

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

      2. unpow2 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in u1 around 0 66.2%

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

      1. +-commutative 66.2%

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

   9. Simplified 66.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1 \cdot u1} \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]

Alternative 8: 88.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
}

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 + (u2 * (u2 * (-19.739208802181317e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

2. Taylor expanded in u2 around 0 89.3%

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

   1. *-rgt-identity 89.3%

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

   2. *-commutative 89.3%

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

   3. associate-*l* 89.3%

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

   4. distribute-lft-out 89.2%

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

   5. unpow2 89.2%

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

4. Simplified 89.2%

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

5. Taylor expanded in u2 around 0 89.2%

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

   1. *-commutative 89.2%

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

   2. unpow2 89.2%

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

   3. associate-*r* 89.2%

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

7. Simplified 89.2%

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

8. Final simplification 89.2%

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

Alternative 9: 88.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
}

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 + ((u2 * u2) * (-19.739208802181317e0)))
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(u2 * u2) * Float32(-19.739208802181317))))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

2. Taylor expanded in u2 around 0 89.3%

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

   1. *-rgt-identity 89.3%

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

   2. *-commutative 89.3%

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

   3. associate-*l* 89.3%

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

   4. distribute-lft-out 89.2%

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

   5. unpow2 89.2%

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

4. Simplified 89.2%

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

5. Final simplification 89.2%

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

Alternative 10: 80.3% 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. Taylor expanded in u2 around 0 80.9%

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

3. Final simplification 80.9%

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

Alternative 11: 63.6% 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. Taylor expanded in u2 around 0 80.9%

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

3. Taylor expanded in u1 around 0 64.9%

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

4. Final simplification 64.9%

   \[\leadsto \sqrt{u1} \]

Alternative 12: 19.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

2. Taylor expanded in u1 around 0 87.7%

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

   1. +-commutative 87.7%

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

   2. unpow2 87.7%

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

   3. fma-udef 87.7%

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

4. Simplified 87.7%

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

5. Taylor expanded in u1 around inf 19.6%

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

   1. *-commutative 19.6%

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

   2. *-commutative 19.6%

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

7. Simplified 19.6%

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

8. Taylor expanded in u2 around 0 19.6%

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

   1. unpow2 19.6%

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

10. Simplified 19.6%

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

11. Final simplification 19.6%

    \[\leadsto u1 + -19.739208802181317 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right) \]

Alternative 13: 19.1% accurate, 209.0× 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 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. Taylor expanded in u1 around 0 87.7%

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

   1. +-commutative 87.7%

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

   2. unpow2 87.7%

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

   3. fma-udef 87.7%

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

4. Simplified 87.7%

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

5. Taylor expanded in u1 around inf 19.6%

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

   1. *-commutative 19.6%

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

   2. *-commutative 19.6%

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

7. Simplified 19.6%

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

8. Taylor expanded in u2 around 0 18.7%

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

9. Final simplification 18.7%

   \[\leadsto u1 \]

Reproduce

herbie shell --seed 2023279 
(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))))