Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 14.0s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin(sqrt((39.47841760436263e0 * (u2 * u2))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-sqr-sqrt 97.7%

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

   2. pow1/2 97.7%

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

   3. pow1/2 97.7%

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

   4. pow-prod-down 98.2%

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

   5. swap-sqr 98.0%

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

   6. metadata-eval 98.6%

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

3. Applied egg-rr 98.6%

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

   1. unpow1/2 98.6%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.003000000026077032)
   (sqrt (* (/ u1 (- 1.0 u1)) (* 39.47841760436263 (* u2 u2))))
   (* (sin (* u2 6.28318530718)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.003000000026077032f) {
		tmp = sqrtf(((u1 / (1.0f - u1)) * (39.47841760436263f * (u2 * u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 + (u1 * u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.003000000026077032e0) then
        tmp = sqrt(((u1 / (1.0e0 - u1)) * (39.47841760436263e0 * (u2 * u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 + (u1 * u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.003000000026077032))
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(Float32(39.47841760436263) * Float32(u2 * u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.003000000026077032))
		tmp = sqrt(((u1 / (single(1.0) - u1)) * (single(39.47841760436263) * (u2 * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00300000003

   1. Initial program 98.3%

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

   2. Taylor expanded in u2 around 0 97.9%

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

      1. add-sqr-sqrt 97.6%

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

      2. sqrt-unprod 97.9%

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

      3. swap-sqr 97.6%

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

      4. metadata-eval 98.1%

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

      5. *-commutative 98.1%

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

      6. *-commutative 98.1%

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

      7. swap-sqr 98.3%

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

      8. add-sqr-sqrt 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in u2 around 0 98.6%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 98.7%

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

      2. associate-*r* 98.6%

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

      3. *-commutative 98.6%

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

      4. associate-*l* 98.7%

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

      5. unpow2 98.7%

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

   9. Simplified 98.7%

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

   if 0.00300000003 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.1%

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

      1. flip-- 98.1%

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

      2. associate-/r/ 98.0%

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

      3. metadata-eval 98.0%

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

      4. +-commutative 98.0%

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

   3. Applied egg-rr 98.0%

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

      1. +-commutative 98.0%

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

      2. distribute-lft-in 98.1%

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

      3. *-rgt-identity 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in u1 around 0 86.0%

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

      1. unpow2 86.0%

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

   8. Simplified 86.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.03500000014901161)
   (sqrt (* (/ u1 (- 1.0 u1)) (* 39.47841760436263 (* u2 u2))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.03500000014901161f) {
		tmp = sqrtf(((u1 / (1.0f - u1)) * (39.47841760436263f * (u2 * u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.03500000014901161e0) then
        tmp = sqrt(((u1 / (1.0e0 - u1)) * (39.47841760436263e0 * (u2 * u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.03500000014901161))
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(Float32(39.47841760436263) * Float32(u2 * u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.03500000014901161))
		tmp = sqrt(((u1 / (single(1.0) - u1)) * (single(39.47841760436263) * (u2 * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0350000001

   1. Initial program 98.2%

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

   2. Taylor expanded in u2 around 0 95.3%

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

      1. add-sqr-sqrt 95.0%

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

      2. sqrt-unprod 95.3%

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

      3. swap-sqr 95.1%

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

      4. metadata-eval 95.5%

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

      5. *-commutative 95.5%

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

      6. *-commutative 95.5%

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

      7. swap-sqr 95.7%

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

      8. add-sqr-sqrt 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. *-commutative 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in u2 around 0 95.9%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 96.0%

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

      2. associate-*r* 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-*l* 96.0%

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

      5. unpow2 96.0%

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

   9. Simplified 96.0%

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

   if 0.0350000001 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.1%

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

   2. Taylor expanded in u1 around 0 78.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Final simplification 98.2%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) / sqrtf(((1.0f / 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 = sin((u2 * 6.28318530718e0)) / sqrt(((1.0e0 / u1) + (-1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. sqrt-div 98.0%

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

   3. associate-*r/ 98.1%

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

3. Applied egg-rr 98.1%

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

   1. associate-/l* 98.3%

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

5. Simplified 98.3%

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

   1. clear-num 98.2%

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

   2. inv-pow 98.2%

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

   3. sqrt-undiv 98.2%

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

7. Applied egg-rr 98.2%

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

   1. unpow-1 98.2%

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

   2. associate-/r/ 98.2%

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

   3. associate-*l/ 98.3%

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

   4. *-lft-identity 98.3%

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

   5. div-sub 98.3%

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

   6. sub-neg 98.3%

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

   7. *-inverses 98.3%

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

   8. metadata-eval 98.3%

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

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. *-commutative 98.2%

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

   2. sqrt-div 98.0%

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

   3. associate-*r/ 98.1%

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

3. Applied egg-rr 98.1%

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

   1. associate-/l* 98.3%

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

5. Simplified 98.3%

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

   1. expm1-log1p-u 96.2%

      \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{1 - u1}}{\sqrt{u1}}\right)\right)}} \]

   2. expm1-udef 96.0%

      \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{1 - u1}}{\sqrt{u1}}\right)} - 1}} \]

   3. sqrt-undiv 96.2%

      \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1 - u1}{u1}}}\right)} - 1} \]

7. Applied egg-rr 96.2%

   \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1 - u1}{u1}}\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1 - u1}{u1}}\right)\right)}} \]

   2. expm1-log1p 98.3%

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

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 7: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 82.4%

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

   1. add-sqr-sqrt 82.1%

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

   2. sqrt-unprod 82.4%

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

   3. swap-sqr 82.2%

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

   4. metadata-eval 82.5%

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

   5. *-commutative 82.5%

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

   6. *-commutative 82.5%

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

   7. swap-sqr 82.6%

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

   8. add-sqr-sqrt 82.9%

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

4. Applied egg-rr 82.9%

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

   1. associate-*l* 82.9%

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

   2. associate-*r/ 82.9%

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

6. Simplified 82.9%

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

7. Final simplification 82.9%

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

Alternative 8: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((u1 / (1.0e0 - u1)) * (39.47841760436263e0 * (u2 * u2))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 82.4%

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

   1. add-sqr-sqrt 82.1%

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

   2. sqrt-unprod 82.4%

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

   3. swap-sqr 82.2%

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

   4. metadata-eval 82.5%

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

   5. *-commutative 82.5%

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

   6. *-commutative 82.5%

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

   7. swap-sqr 82.6%

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

   8. add-sqr-sqrt 82.9%

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

4. Applied egg-rr 82.9%

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

   1. *-commutative 82.9%

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

6. Simplified 82.9%

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

7. Taylor expanded in u2 around 0 82.8%

   \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]
8. Step-by-step derivation

   1. associate-*l/ 82.9%

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

   2. associate-*r* 82.9%

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

   3. *-commutative 82.9%

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

   4. associate-*l* 82.9%

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

   5. unpow2 82.9%

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

9. Simplified 82.9%

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

10. Final simplification 82.9%

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

Alternative 9: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 82.4%

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

3. Final simplification 82.4%

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

Alternative 10: 81.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 82.4%

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

   1. clear-num 82.3%

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

   2. sqrt-div 82.3%

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

   3. metadata-eval 82.3%

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

   4. div-sub 82.2%

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

   5. *-inverses 82.2%

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

   6. sub-neg 82.2%

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

   7. metadata-eval 82.2%

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

4. Applied egg-rr 82.2%

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

   1. associate-*l/ 82.4%

      \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{1 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}} \]

   2. *-un-lft-identity 82.4%

      \[\leadsto 6.28318530718 \cdot \frac{\color{blue}{u2}}{\sqrt{\frac{1}{u1} + -1}} \]

6. Applied egg-rr 82.4%

   \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{u2}{\sqrt{\frac{1}{u1} + -1}}} \]

7. Final simplification 82.4%

   \[\leadsto 6.28318530718 \cdot \frac{u2}{\sqrt{\frac{1}{u1} + -1}} \]

Alternative 11: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{\frac{u1 \cdot 39.47841760436263}{1 - u1}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 82.4%

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

   1. add-sqr-sqrt 82.1%

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

   2. sqrt-unprod 82.4%

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

   3. swap-sqr 82.2%

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

   4. metadata-eval 82.5%

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

   5. *-commutative 82.5%

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

   6. *-commutative 82.5%

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

   7. swap-sqr 82.6%

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

   8. add-sqr-sqrt 82.9%

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

4. Applied egg-rr 82.9%

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

   1. *-commutative 82.9%

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

6. Simplified 82.9%

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

   1. associate-*r* 82.9%

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

   2. sqrt-prod 82.8%

      \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u1}{1 - u1}} \cdot \sqrt{u2 \cdot u2}} \]

   3. sqrt-prod 82.2%

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

   4. add-sqr-sqrt 82.8%

      \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u1}{1 - u1}} \cdot \color{blue}{u2} \]

8. Applied egg-rr 82.8%

   \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u1}{1 - u1}} \cdot u2} \]
9. Step-by-step derivation

   1. unpow1/2 82.8%

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

   2. *-commutative 82.8%

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

   3. unpow1/2 82.8%

      \[\leadsto u2 \cdot \color{blue}{\sqrt{39.47841760436263 \cdot \frac{u1}{1 - u1}}} \]

   4. associate-*r/ 82.8%

      \[\leadsto u2 \cdot \sqrt{\color{blue}{\frac{39.47841760436263 \cdot u1}{1 - u1}}} \]

10. Simplified 82.8%

    \[\leadsto \color{blue}{u2 \cdot \sqrt{\frac{39.47841760436263 \cdot u1}{1 - u1}}} \]

11. Final simplification 82.8%

    \[\leadsto u2 \cdot \sqrt{\frac{u1 \cdot 39.47841760436263}{1 - u1}} \]

Alternative 12: 64.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * 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 = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(u1)))
end

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 82.4%

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

3. Taylor expanded in u1 around 0 64.3%

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

4. Final simplification 64.3%

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

Alternative 13: -0.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{-39.47841760436263} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf(-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 = u2 * sqrt((-39.47841760436263e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 82.4%

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

   1. add-sqr-sqrt 82.1%

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

   2. sqrt-unprod 82.4%

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

   3. swap-sqr 82.2%

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

   4. metadata-eval 82.5%

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

   5. *-commutative 82.5%

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

   6. *-commutative 82.5%

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

   7. swap-sqr 82.6%

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

   8. add-sqr-sqrt 82.9%

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

4. Applied egg-rr 82.9%

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

   1. *-commutative 82.9%

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

6. Simplified 82.9%

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

7. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto \color{blue}{u2 \cdot \sqrt{-39.47841760436263}} \]

8. Final simplification -0.0%

   \[\leadsto u2 \cdot \sqrt{-39.47841760436263} \]

Reproduce

herbie shell --seed 2023272 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :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))) (sin (* 6.28318530718 u2))))