Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 14.1s

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

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((1.0e0 - u1) / u1) ** (-0.5e0)) * cos((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) - u1) / u1) ^ single(-0.5)) * 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. Step-by-step derivation

   1. pow1/2 98.9%

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

   2. clear-num 98.9%

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

   3. inv-pow 98.9%

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

   4. pow-pow 98.9%

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

   5. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

Alternative 2: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999979734420776:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999979734420776)
     (* t_0 (sqrt (* u1 (+ 1.0 u1))))
     (pow (/ (- 1.0 u1) 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 <= 0.9999979734420776f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + u1)));
	} else {
		tmp = powf(((1.0f - u1) / 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 <= 0.9999979734420776e0) then
        tmp = t_0 * sqrt((u1 * (1.0e0 + u1)))
    else
        tmp = ((1.0e0 - u1) / 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(0.9999979734420776))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	else
		tmp = Float32(Float32(Float32(1.0) - u1) / 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(0.9999979734420776))
		tmp = t_0 * sqrt((u1 * (single(1.0) + u1)));
	else
		tmp = ((single(1.0) - u1) / 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)) < 0.999997973

   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 89.0%

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

   if 0.999997973 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.5%

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

      1. pow1/2 99.5%

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

      2. clear-num 99.3%

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

      3. associate-/r/ 99.0%

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

      4. unpow-prod-down 98.7%

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

      5. inv-pow 98.7%

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

      6. pow-pow 98.7%

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

      7. metadata-eval 98.7%

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

      8. pow1/2 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in u2 around 0 98.6%

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

      1. pow1/2 99.5%

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

      2. clear-num 99.3%

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

      3. inv-pow 99.3%

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

      4. pow-pow 99.5%

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

      5. metadata-eval 99.5%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999979734420776:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999960064888:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

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

   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 78.6%

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

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

   1. Initial program 99.4%

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

      1. pow1/2 99.4%

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

      2. clear-num 99.3%

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

      3. associate-/r/ 99.0%

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

      4. unpow-prod-down 98.7%

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

      5. inv-pow 98.7%

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

      6. pow-pow 98.7%

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

      7. metadata-eval 98.7%

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

      8. pow1/2 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in u2 around 0 97.9%

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

      1. pow1/2 99.4%

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

      2. clear-num 99.3%

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

      3. inv-pow 99.3%

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

      4. pow-pow 99.5%

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

      5. metadata-eval 99.5%

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999960064888:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1 - u1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 5: 80.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(1.0) - u1) / 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%

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

   2. clear-num 98.9%

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

   3. associate-/r/ 98.6%

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

   4. unpow-prod-down 98.3%

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

   5. inv-pow 98.3%

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

   6. pow-pow 98.3%

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

   7. metadata-eval 98.3%

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

   8. pow1/2 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0 75.7%

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

   1. pow1/2 98.9%

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

   2. clear-num 98.9%

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

   3. inv-pow 98.9%

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

   4. pow-pow 98.9%

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

   5. metadata-eval 98.9%

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

7. Applied egg-rr 75.8%

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

Alternative 6: 80.4% 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. Step-by-step derivation

   1. pow1/2 98.9%

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

   2. clear-num 98.9%

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

   3. associate-/r/ 98.6%

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

   4. unpow-prod-down 98.3%

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

   5. inv-pow 98.3%

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

   6. pow-pow 98.3%

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

   7. metadata-eval 98.3%

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

   8. pow1/2 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0 75.7%

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

Alternative 7: 71.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

   1. pow1/2 98.9%

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

   2. clear-num 98.9%

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

   3. associate-/r/ 98.6%

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

   4. unpow-prod-down 98.3%

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

   5. inv-pow 98.3%

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

   6. pow-pow 98.3%

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

   7. metadata-eval 98.3%

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

   8. pow1/2 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0 75.7%

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

6. Taylor expanded in u1 around 0 67.4%

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

Alternative 8: 63.2% 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. Step-by-step derivation

   1. pow1/2 98.9%

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

   2. clear-num 98.9%

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

   3. associate-/r/ 98.6%

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

   4. unpow-prod-down 98.3%

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

   5. inv-pow 98.3%

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

   6. pow-pow 98.3%

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

   7. metadata-eval 98.3%

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

   8. pow1/2 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0 75.7%

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

6. Taylor expanded in u1 around 0 60.1%

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

Reproduce

herbie shell --seed 2024085 
(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))))