Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 10.4s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u1 around inf 98.7%

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

   1. sub-neg 98.7%

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

   2. metadata-eval 98.7%

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

   3. distribute-rgt-in 98.9%

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

   4. neg-mul-1 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999954104423523:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999954104423523)
     (* t_0 (sqrt u1))
     (cbrt (pow (/ u1 (- 1.0 u1)) 1.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999954104423523f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = cbrtf(powf((u1 / (1.0f - u1)), 1.5f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999954104423523))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = cbrt((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99999541

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 78.9%

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

   if 0.99999541 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0 98.0%

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

      1. add-cbrt-cube 98.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \]

      2. pow1/3 94.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 95.0%

         \[\leadsto {\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \]

      4. pow1 95.0%

         \[\leadsto {\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \]

      5. pow1/2 95.0%

         \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 95.0%

         \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 95.0%

         \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 95.0%

      \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 98.1%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999954104423523:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999954104423523:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999954104423523)
     (* t_0 (sqrt u1))
     (pow (/ u1 (- 1.0 u1)) 0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999954104423523f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf((u1 / (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) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999954104423523e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = (u1 / (1.0e0 - u1)) ** 0.5e0
    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(0.9999954104423523))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5);
	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(0.9999954104423523))
		tmp = t_0 * sqrt(u1);
	else
		tmp = (u1 / (single(1.0) - u1)) ^ single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99999541

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 78.9%

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

   if 0.99999541 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0 98.0%

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

      1. pow1/2 99.3%

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

   5. Applied egg-rr 98.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999954104423523:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.000859999970998615:\\ \;\;\;\;\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.000859999970998615)
   (cbrt (pow (/ u1 (- 1.0 u1)) 1.5))
   (* (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) <= 0.000859999970998615f) {
		tmp = cbrtf(powf((u1 / (1.0f - u1)), 1.5f));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.000859999970998615))
		tmp = cbrt((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 8.59999971e-4

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0 98.9%

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

      1. add-cbrt-cube 98.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \]

      2. pow1/3 95.6%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 95.7%

         \[\leadsto {\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \]

      4. pow1 95.7%

         \[\leadsto {\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}^{0.3333333333333333} \]

      5. pow1/2 95.7%

         \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 95.6%

         \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 95.6%

         \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   5. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   6. Step-by-step derivation

      1. unpow1/3 99.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \]

   7. Simplified 99.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}} \]

   if 8.59999971e-4 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0 89.7%

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

      1. +-commutative 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.000859999970998615:\\ \;\;\;\;\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((6.28318530718f * u2)) * powf((u1 / (1.0f - u1)), 0.5f);
}

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)) * ((u1 / (1.0e0 - u1)) ** 0.5e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. pow1/2 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 6: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = cos((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Final simplification 98.9%

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

Alternative 7: 79.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((u1 / (1.0f - u1)), 0.5f);
}

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 - u1)) ** 0.5e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 79.5%

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

   1. pow1/2 98.9%

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

5. Applied egg-rr 79.6%

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

Alternative 8: 79.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 79.5%

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

Alternative 9: 71.1% 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 98.9%

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

3. Taylor expanded in u2 around 0 79.5%

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

4. Taylor expanded in u1 around 0 72.3%

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

   1. +-commutative 88.5%

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

6. Simplified 72.3%

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

Alternative 10: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {u1}^{0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow u1 0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(u1, 0.5f);
}

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 ** 0.5e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return u1 ^ Float32(0.5)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 ^ single(0.5);
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 79.5%

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

4. Taylor expanded in u1 around 0 63.6%

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

   1. pow1/2 63.6%

      \[\leadsto \color{blue}{{u1}^{0.5}} \]

6. Applied egg-rr 63.6%

   \[\leadsto \color{blue}{{u1}^{0.5}} \]
7. Add Preprocessing

Alternative 11: 62.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 79.5%

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

4. Taylor expanded in u1 around 0 63.6%

   \[\leadsto \sqrt{\color{blue}{u1}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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))))