Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.7s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. *-commutative 98.4%

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

   2. sqrt-div 98.1%

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

   3. associate-*r/ 98.0%

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

3. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]
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 96.4%

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

   2. expm1-udef 96.2%

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

   3. sqrt-undiv 96.3%

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

7. Applied egg-rr 96.3%

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

   1. expm1-def 96.4%

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

   2. expm1-log1p 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 2: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02319999970495701:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u1 \cdot \left(u2 \cdot u2\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.02319999970495701)
   (sqrt (* 39.47841760436263 (/ (* u1 (* u2 u2)) (- 1.0 u1))))
   (* (sin (* 6.28318530718 u2)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.02319999970495701f) {
		tmp = sqrtf((39.47841760436263f * ((u1 * (u2 * u2)) / (1.0f - u1))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * 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 ((6.28318530718e0 * u2) <= 0.02319999970495701e0) then
        tmp = sqrt((39.47841760436263e0 * ((u1 * (u2 * u2)) / (1.0e0 - u1))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 + (u1 * u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.02319999970495701))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(Float32(u1 * Float32(u2 * u2)) / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.02319999970495701))
		tmp = sqrt((single(39.47841760436263) * ((u1 * (u2 * u2)) / (single(1.0) - u1))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0231999997

   1. Initial program 98.5%

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

      1. add-sqr-sqrt 98.0%

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

      2. pow1/2 98.0%

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

      3. pow1/2 98.0%

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

      4. pow-prod-down 98.5%

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

      5. swap-sqr 98.4%

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

      6. add-sqr-sqrt 98.6%

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

      7. pow2 98.6%

         \[\leadsto {\left(\frac{u1}{1 - u1} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{2}}\right)}^{0.5} \]

   3. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{2}\right)}^{0.5}} \]
   4. Step-by-step derivation

      1. unpow1/2 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in u2 around 0 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(-519.5151521814831 \cdot {u2}^{4} + 39.47841760436263 \cdot {u2}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(39.47841760436263 \cdot {u2}^{2} + -519.5151521814831 \cdot {u2}^{4}\right)}} \]

      2. fma-def 99.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\mathsf{fma}\left(39.47841760436263, {u2}^{2}, -519.5151521814831 \cdot {u2}^{4}\right)}} \]

      3. unpow2 99.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \mathsf{fma}\left(39.47841760436263, \color{blue}{u2 \cdot u2}, -519.5151521814831 \cdot {u2}^{4}\right)} \]

   8. Simplified 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\mathsf{fma}\left(39.47841760436263, u2 \cdot u2, -519.5151521814831 \cdot {u2}^{4}\right)}} \]

   9. Taylor expanded in u2 around 0 97.1%

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

       1. *-commutative 97.1%

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

       2. unpow2 97.1%

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

   11. Simplified 97.1%

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

   if 0.0231999997 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 98.0%

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

      3. *-rgt-identity 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in u1 around 0 88.1%

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

      1. +-commutative 88.1%

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

      2. unpow2 88.1%

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

   8. Simplified 88.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02319999970495701:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u1 \cdot \left(u2 \cdot u2\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02319999970495701:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u1 \cdot \left(u2 \cdot u2\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.02319999970495701)
   (sqrt (* 39.47841760436263 (/ (* u1 (* u2 u2)) (- 1.0 u1))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.02319999970495701f) {
		tmp = sqrtf((39.47841760436263f * ((u1 * (u2 * u2)) / (1.0f - u1))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * 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 ((6.28318530718e0 * u2) <= 0.02319999970495701e0) then
        tmp = sqrt((39.47841760436263e0 * ((u1 * (u2 * u2)) / (1.0e0 - u1))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.02319999970495701))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(Float32(u1 * Float32(u2 * u2)) / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.02319999970495701))
		tmp = sqrt((single(39.47841760436263) * ((u1 * (u2 * u2)) / (single(1.0) - u1))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0231999997

   1. Initial program 98.5%

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

      1. add-sqr-sqrt 98.0%

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

      2. pow1/2 98.0%

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

      3. pow1/2 98.0%

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

      4. pow-prod-down 98.5%

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

      5. swap-sqr 98.4%

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

      6. add-sqr-sqrt 98.6%

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

      7. pow2 98.6%

         \[\leadsto {\left(\frac{u1}{1 - u1} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{2}}\right)}^{0.5} \]

   3. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{2}\right)}^{0.5}} \]
   4. Step-by-step derivation

      1. unpow1/2 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in u2 around 0 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(-519.5151521814831 \cdot {u2}^{4} + 39.47841760436263 \cdot {u2}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(39.47841760436263 \cdot {u2}^{2} + -519.5151521814831 \cdot {u2}^{4}\right)}} \]

      2. fma-def 99.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\mathsf{fma}\left(39.47841760436263, {u2}^{2}, -519.5151521814831 \cdot {u2}^{4}\right)}} \]

      3. unpow2 99.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \mathsf{fma}\left(39.47841760436263, \color{blue}{u2 \cdot u2}, -519.5151521814831 \cdot {u2}^{4}\right)} \]

   8. Simplified 99.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\mathsf{fma}\left(39.47841760436263, u2 \cdot u2, -519.5151521814831 \cdot {u2}^{4}\right)}} \]

   9. Taylor expanded in u2 around 0 97.1%

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

       1. *-commutative 97.1%

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

       2. unpow2 97.1%

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

   11. Simplified 97.1%

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

   if 0.0231999997 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.0%

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

   2. Taylor expanded in u1 around 0 75.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02319999970495701:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u1 \cdot \left(u2 \cdot u2\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Final simplification 98.4%

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

Alternative 5: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. add-sqr-sqrt 94.5%

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

   2. pow1/2 94.5%

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

   3. pow1/2 94.5%

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

   4. pow-prod-down 95.2%

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

   5. swap-sqr 95.1%

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

   6. add-sqr-sqrt 95.2%

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

   7. pow2 95.2%

      \[\leadsto {\left(\frac{u1}{1 - u1} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{2}}\right)}^{0.5} \]

3. Applied egg-rr 95.2%

   \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{2}\right)}^{0.5}} \]
4. Step-by-step derivation

   1. unpow1/2 95.2%

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

5. Simplified 95.2%

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

6. Taylor expanded in u2 around 0 84.9%

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

   1. *-commutative 84.9%

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

   2. associate-*l/ 84.8%

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

   3. unpow2 84.8%

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

8. Simplified 84.8%

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

9. Final simplification 84.8%

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

Alternative 6: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. add-sqr-sqrt 94.5%

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

   2. pow1/2 94.5%

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

   3. pow1/2 94.5%

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

   4. pow-prod-down 95.2%

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

   5. swap-sqr 95.1%

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

   6. add-sqr-sqrt 95.2%

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

   7. pow2 95.2%

      \[\leadsto {\left(\frac{u1}{1 - u1} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{2}}\right)}^{0.5} \]

3. Applied egg-rr 95.2%

   \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{2}\right)}^{0.5}} \]
4. Step-by-step derivation

   1. unpow1/2 95.2%

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

5. Simplified 95.2%

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

6. Taylor expanded in u2 around 0 90.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(-519.5151521814831 \cdot {u2}^{4} + 39.47841760436263 \cdot {u2}^{2}\right)}} \]
7. Step-by-step derivation

   1. +-commutative 90.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(39.47841760436263 \cdot {u2}^{2} + -519.5151521814831 \cdot {u2}^{4}\right)}} \]

   2. fma-def 90.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\mathsf{fma}\left(39.47841760436263, {u2}^{2}, -519.5151521814831 \cdot {u2}^{4}\right)}} \]

   3. unpow2 90.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \mathsf{fma}\left(39.47841760436263, \color{blue}{u2 \cdot u2}, -519.5151521814831 \cdot {u2}^{4}\right)} \]

8. Simplified 90.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\mathsf{fma}\left(39.47841760436263, u2 \cdot u2, -519.5151521814831 \cdot {u2}^{4}\right)}} \]

9. Taylor expanded in u2 around 0 84.9%

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

    1. *-commutative 84.9%

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

    2. unpow2 84.9%

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

11. Simplified 84.9%

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

12. Final simplification 84.9%

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

Alternative 7: 81.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 84.6%

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

3. Final simplification 84.6%

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

Alternative 8: 64.1% 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 98.4%

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

2. Taylor expanded in u2 around 0 84.6%

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

3. Taylor expanded in u1 around 0 66.3%

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

4. Final simplification 66.3%

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

Reproduce

herbie shell --seed 2023178 
(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))))