Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 14.5s

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf(sqrtf((powf(u2, 2.0f) * 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 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
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. 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. sqrt-unprod 98.9%

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

   3. *-commutative 98.9%

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

   4. *-commutative 98.9%

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

   5. swap-sqr 98.9%

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

   6. pow2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{{u2}^{2}} \cdot \left(6.28318530718 \cdot 6.28318530718\right)}\right) \]

   7. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

5. Final simplification 99.0%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \]
6. Add Preprocessing

Alternative 2: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sqrt{\frac{1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (* (sqrt u1) (sqrt (/ 1.0 (- 1.0 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(u1) * sqrtf((1.0f / (1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = cos((u2 * 6.28318530718e0))
    if (t_0 <= 0.9999499917030334e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = sqrt(u1) * sqrt((1.0e0 / (1.0e0 - u1)))
    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(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(sqrt(u1) * sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - u1))));
	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(0.9999499917030334))
		tmp = t_0 * sqrt(u1);
	else
		tmp = sqrt(u1) * sqrt((single(1.0) / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999949992

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 73.7%

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

   if 0.999949992 < (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. Taylor expanded in u2 around 0 96.5%

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

      1. pow1/2 96.5%

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

      2. div-inv 96.2%

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

      3. unpow-prod-down 96.0%

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

      4. pow1/2 96.0%

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

   5. Applied egg-rr 96.0%

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

      1. unpow1/2 96.0%

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

   7. Simplified 96.0%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sqrt{\frac{1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (/ 1.0 (/ (sqrt (- 1.0 u1)) (sqrt u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = 1.0f / (sqrtf((1.0f - u1)) / sqrtf(u1));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = cos((u2 * 6.28318530718e0))
    if (t_0 <= 0.9999499917030334e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = 1.0e0 / (sqrt((1.0e0 - u1)) / sqrt(u1))
    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(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(Float32(1.0) / Float32(sqrt(Float32(Float32(1.0) - u1)) / sqrt(u1)));
	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(0.9999499917030334))
		tmp = t_0 * sqrt(u1);
	else
		tmp = single(1.0) / (sqrt((single(1.0) - u1)) / sqrt(u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999949992

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 73.7%

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

   if 0.999949992 < (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. Taylor expanded in u2 around 0 96.5%

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

      1. sqrt-div 96.0%

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

      2. clear-num 95.7%

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

   5. Applied egg-rr 95.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{u1}{1 - u1}\right)}^{0.16666666666666666}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (pow (pow (/ u1 (- 1.0 u1)) 0.16666666666666666) 3.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf(powf((u1 / (1.0f - u1)), 0.16666666666666666f), 3.0f);
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = cos((u2 * 6.28318530718e0))
    if (t_0 <= 0.9999499917030334e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = ((u1 / (1.0e0 - u1)) ** 0.16666666666666666e0) ** 3.0e0
    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(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = (Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.16666666666666666)) ^ Float32(3.0);
	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(0.9999499917030334))
		tmp = t_0 * sqrt(u1);
	else
		tmp = ((u1 / (single(1.0) - u1)) ^ single(0.16666666666666666)) ^ single(3.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999949992

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 73.7%

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

   if 0.999949992 < (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. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 99.4%

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

      3. *-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. swap-sqr 99.4%

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

      6. pow2 99.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{{u2}^{2}} \cdot \left(6.28318530718 \cdot 6.28318530718\right)}\right) \]

      7. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \]
   5. Step-by-step derivation

      1. add-cube-cbrt 97.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right) \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}} \]

      2. pow3 97.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right)}^{3}} \]

      3. *-commutative 97.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{39.47841760436263 \cdot {u2}^{2}}}\right)}\right)}^{3} \]

      4. sqrt-prod 97.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263} \cdot \sqrt{{u2}^{2}}\right)}}\right)}^{3} \]

      5. metadata-eval 97.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{6.28318530718} \cdot \sqrt{{u2}^{2}}\right)}\right)}^{3} \]

      6. sqrt-pow1 97.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot \color{blue}{{u2}^{\left(\frac{2}{2}\right)}}\right)}\right)}^{3} \]

      7. metadata-eval 97.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot {u2}^{\color{blue}{1}}\right)}\right)}^{3} \]

      8. pow1 97.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot \color{blue}{u2}\right)}\right)}^{3} \]

      9. *-commutative 97.4%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(u2 \cdot 6.28318530718\right)}}\right)}^{3} \]

   6. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(u2 \cdot 6.28318530718\right)}\right)}^{3}} \]

   7. Taylor expanded in u2 around 0 93.4%

      \[\leadsto {\color{blue}{\left({\left(\frac{1 \cdot u1}{1 - u1}\right)}^{0.16666666666666666}\right)}}^{3} \]
   8. Step-by-step derivation

      1. *-lft-identity 93.4%

         \[\leadsto {\left({\left(\frac{\color{blue}{u1}}{1 - u1}\right)}^{0.16666666666666666}\right)}^{3} \]

   9. Simplified 93.4%

      \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{0.16666666666666666}\right)}}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{u1}{1 - u1}\right)}^{0.16666666666666666}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{u1}{1 - u1}\right)}^{0.25}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (pow (pow (/ u1 (- 1.0 u1)) 0.25) 2.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf(powf((u1 / (1.0f - u1)), 0.25f), 2.0f);
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = cos((u2 * 6.28318530718e0))
    if (t_0 <= 0.9999499917030334e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = ((u1 / (1.0e0 - u1)) ** 0.25e0) ** 2.0e0
    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(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = (Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.25)) ^ Float32(2.0);
	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(0.9999499917030334))
		tmp = t_0 * sqrt(u1);
	else
		tmp = ((u1 / (single(1.0) - u1)) ^ single(0.25)) ^ single(2.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999949992

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 73.7%

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

   if 0.999949992 < (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. Taylor expanded in u2 around 0 96.5%

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

      1. add-sqr-sqrt 95.4%

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

      2. pow2 95.5%

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

      3. pow1/2 95.5%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}}\right)}^{2} \]

      4. sqrt-pow1 95.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \]

      5. metadata-eval 95.5%

         \[\leadsto {\left({\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{0.25}}\right)}^{2} \]

   5. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{0.25}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{u1}{1 - u1}\right)}^{0.25}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (pow (pow (/ u1 (- 1.0 u1)) 1.5) 0.3333333333333333))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = powf(powf((u1 / (1.0f - u1)), 1.5f), 0.3333333333333333f);
	}
	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 <= 0.9999499917030334e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = ((u1 / (1.0e0 - u1)) ** 1.5e0) ** 0.3333333333333333e0
    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(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = (Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)) ^ Float32(0.3333333333333333);
	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(0.9999499917030334))
		tmp = t_0 * sqrt(u1);
	else
		tmp = ((u1 / (single(1.0) - u1)) ^ single(1.5)) ^ single(0.3333333333333333);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999949992

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 73.7%

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

   if 0.999949992 < (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. Taylor expanded in u2 around 0 96.5%

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

      1. add-cbrt-cube 96.5%

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

      2. pow1/3 93.3%

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

      3. add-sqr-sqrt 93.4%

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

      4. pow1 93.4%

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

      5. pow1/2 93.4%

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

      6. pow-prod-up 93.3%

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

      7. metadata-eval 93.3%

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

   5. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{u1}{1 - u1}\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.006000000052154064)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (cos (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.006000000052154064f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * 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) <= 0.006000000052154064e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * 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(0.006000000052154064))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * 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(0.006000000052154064))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00600000005

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0 97.3%

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

   if 0.00600000005 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0 85.5%

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

      1. +-commutative 43.7%

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

   5. Simplified 85.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(-1.0) + (single(1.0) / u1)) ^ single(-0.5)) * cos((u2 * single(6.28318530718)));
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. 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. sqrt-unprod 98.9%

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

   3. *-commutative 98.9%

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

   4. *-commutative 98.9%

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

   5. swap-sqr 98.9%

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

   6. pow2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{{u2}^{2}} \cdot \left(6.28318530718 \cdot 6.28318530718\right)}\right) \]

   7. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \]
5. Step-by-step derivation

   1. add-cube-cbrt 97.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right) \cdot \sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}} \]

   2. pow3 97.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}\right)}^{3}} \]

   3. *-commutative 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{39.47841760436263 \cdot {u2}^{2}}}\right)}\right)}^{3} \]

   4. sqrt-prod 97.1%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263} \cdot \sqrt{{u2}^{2}}\right)}}\right)}^{3} \]

   5. metadata-eval 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{6.28318530718} \cdot \sqrt{{u2}^{2}}\right)}\right)}^{3} \]

   6. sqrt-pow1 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot \color{blue}{{u2}^{\left(\frac{2}{2}\right)}}\right)}\right)}^{3} \]

   7. metadata-eval 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot {u2}^{\color{blue}{1}}\right)}\right)}^{3} \]

   8. pow1 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot \color{blue}{u2}\right)}\right)}^{3} \]

   9. *-commutative 97.2%

      \[\leadsto {\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(u2 \cdot 6.28318530718\right)}}\right)}^{3} \]

6. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(u2 \cdot 6.28318530718\right)}\right)}^{3}} \]
7. Step-by-step derivation

   1. rem-cube-cbrt 98.9%

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

   2. add-sqr-sqrt 97.8%

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

   3. metadata-eval 97.9%

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

   4. unpow1/2 97.9%

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

   5. *-commutative 97.9%

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

   6. associate-*l* 97.8%

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

   7. pow1/2 97.8%

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

   8. sqrt-pow1 98.1%

      \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - u1}}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot u2\right)\right) \]

   9. metadata-eval 98.1%

      \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{\frac{u1}{1 - u1}}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot u2\right)\right) \]

   10. pow1/2 98.2%

       \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left(\sqrt{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot u2\right)\right) \]

   11. sqrt-pow1 97.9%

       \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot u2\right)\right) \]

   12. metadata-eval 97.9%

       \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left({\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{0.25}} \cdot \cos \left({39.47841760436263}^{0.5} \cdot u2\right)\right) \]

   13. unpow1/2 97.9%

       \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left({\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \cos \left(\color{blue}{\sqrt{39.47841760436263}} \cdot u2\right)\right) \]

   14. metadata-eval 97.9%

       \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left({\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \cos \left(\color{blue}{6.28318530718} \cdot u2\right)\right) \]

8. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left({\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \cos \left(6.28318530718 \cdot u2\right)\right)} \]
9. Step-by-step derivation

   1. pow1 97.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \left({\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot \cos \left(6.28318530718 \cdot u2\right)\right)\right)}^{1}} \]

   2. associate-*r* 98.0%

      \[\leadsto {\color{blue}{\left(\left({\left(\frac{u1}{1 - u1}\right)}^{0.25} \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.25}\right) \cdot \cos \left(6.28318530718 \cdot u2\right)\right)}}^{1} \]

   3. pow-prod-up 99.0%

      \[\leadsto {\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(0.25 + 0.25\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right)\right)}^{1} \]

   4. metadata-eval 99.0%

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

   5. pow1/2 98.9%

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

   6. *-commutative 98.9%

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

   7. clear-num 98.8%

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

   8. unpow-1 98.8%

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

   9. sqrt-pow1 99.0%

      \[\leadsto {\left(\cos \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(\frac{-1}{2}\right)}}\right)}^{1} \]

   10. div-sub 99.0%

       \[\leadsto {\left(\cos \left(6.28318530718 \cdot u2\right) \cdot {\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{1} \]

   11. *-inverses 99.0%

       \[\leadsto {\left(\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1} - \color{blue}{1}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{1} \]

   12. sub-neg 99.0%

       \[\leadsto {\left(\cos \left(6.28318530718 \cdot u2\right) \cdot {\color{blue}{\left(\frac{1}{u1} + \left(-1\right)\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{1} \]

   13. metadata-eval 99.0%

       \[\leadsto {\left(\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1} + \color{blue}{-1}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{1} \]

   14. metadata-eval 99.0%

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

10. Applied egg-rr 99.0%

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

    1. unpow1 99.0%

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

    2. *-commutative 99.0%

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

    3. +-commutative 99.0%

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

12. Simplified 99.0%

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

13. Final simplification 99.0%

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

Alternative 9: 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 98.9%

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

3. Final simplification 98.9%

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

Alternative 10: 71.9% 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 79.7%

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

4. Taylor expanded in u1 around 0 72.5%

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

   1. +-commutative 72.5%

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

6. Simplified 72.5%

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

7. Final simplification 72.5%

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

Alternative 11: 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 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 79.7%

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

4. Final simplification 79.7%

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

Alternative 12: 63.4% 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 79.7%

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

4. Taylor expanded in u1 around 0 64.7%

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

5. Final simplification 64.7%

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

Alternative 13: 20.2% accurate, 29.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* u1 (+ 1.0 (/ 0.5 u1))))

C

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

Fortran

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) / u1)))
end

MATLAB

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

Derivation

1. Initial program 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 79.7%

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

4. Taylor expanded in u1 around 0 72.5%

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

   1. +-commutative 72.5%

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

6. Simplified 72.5%

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

7. Taylor expanded in u1 around inf 19.9%

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

   1. associate-*r/ 19.9%

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

   2. metadata-eval 19.9%

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

9. Simplified 19.9%

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

10. Final simplification 19.9%

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

Reproduce

herbie shell --seed 2024066 
(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))))