Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.9s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

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

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

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

Derivation

1. Initial program 98.4%

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

Alternative 2: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.15000000596046448)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.15000000596046448f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * 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 ((6.28318530718e0 * u2) <= 0.15000000596046448e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.15000000596046448))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * 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 ((single(6.28318530718) * u2) <= single(0.15000000596046448))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sin((single(6.28318530718) * u2)) * 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) < 0.150000006

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0 98.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 98.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   5. Applied egg-rr 98.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   if 0.150000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.0%

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

   3. Taylor expanded in u1 around 0 82.1%

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

      1. +-commutative 82.1%

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

   5. Simplified 82.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.5)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sin (* 6.28318530718 u2)) (pow (/ 1.0 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.5f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * powf((1.0f / u1), -0.5f);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.5e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = sin((6.28318530718e0 * u2)) * ((1.0e0 / u1) ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.5))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * (Float32(Float32(1.0) / u1) ^ Float32(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.5))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sin((single(6.28318530718) * u2)) * ((single(1.0) / u1) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.5

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0 96.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 96.7%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   5. Applied egg-rr 96.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   if 0.5 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.0%

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

      1. add-sqr-sqrt 66.6%

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

      2. sqrt-unprod 68.0%

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

      3. swap-sqr 68.0%

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

      4. add-sqr-sqrt 68.0%

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

      5. pow2 68.0%

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

   4. Applied egg-rr 68.0%

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

      1. sqrt-prod 68.0%

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

      2. clear-num 68.2%

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

      3. sqrt-div 67.9%

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

      4. metadata-eval 67.9%

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

      5. pow1/2 67.9%

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

      6. pow-flip 68.3%

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

      7. div-sub 68.3%

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

      8. *-inverses 68.3%

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

      9. sub-neg 68.3%

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

      10. metadata-eval 68.3%

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

      11. metadata-eval 68.3%

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

      12. sqrt-pow1 96.1%

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

      13. metadata-eval 96.1%

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

      14. pow1 96.1%

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

      15. *-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in u1 around 0 71.2%

      \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.5)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.5f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} 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) <= 0.5e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    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(0.5))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	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(0.5))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.5

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0 96.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 96.7%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   5. Applied egg-rr 96.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

   if 0.5 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.0%

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

   3. Taylor expanded in u1 around 0 71.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (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))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 90.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot {u2}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 90.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]

5. Applied egg-rr 90.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)}\right)\right) \]
6. Add Preprocessing

Alternative 6: 81.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 83.0%

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

   1. *-commutative 83.0%

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

   2. associate-*r* 83.1%

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

5. Simplified 83.1%

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

6. Final simplification 83.1%

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

Alternative 7: 81.9% 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 98.4%

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

3. Taylor expanded in u2 around 0 83.0%

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

Alternative 8: 73.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 83.0%

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

4. Taylor expanded in u1 around 0 74.9%

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

   1. +-commutative 86.4%

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

6. Simplified 74.9%

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

7. Final simplification 74.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\right) \]
8. Add Preprocessing

Alternative 9: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 83.0%

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

4. Taylor expanded in u1 around 0 65.1%

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

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

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

   1. *-commutative -0.0%

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

   2. associate-*r* -0.0%

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

   3. *-commutative -0.0%

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

   4. unpow2 -0.0%

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

   5. rem-square-sqrt 65.1%

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

   6. associate-*l* 65.1%

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

   7. metadata-eval 65.1%

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

   8. associate-*l* 65.2%

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

7. Simplified 65.2%

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

   1. add-sqr-sqrt 65.1%

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

   2. sqrt-unprod 65.2%

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

   3. swap-sqr 65.2%

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

   4. add-sqr-sqrt 65.2%

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

   5. metadata-eval 65.2%

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

9. Applied egg-rr 65.2%

   \[\leadsto u2 \cdot \color{blue}{\sqrt{u1 \cdot 39.47841760436263}} \]
10. Add Preprocessing

Alternative 10: 64.7% 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. Add Preprocessing

3. Taylor expanded in u2 around 0 83.0%

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

4. Taylor expanded in u1 around 0 65.1%

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

5. Final simplification 65.1%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Add Preprocessing

Alternative 11: 4.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 83.0%

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

4. Taylor expanded in u1 around 0 65.1%

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

   1. add-sqr-sqrt 65.1%

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

   2. sqrt-unprod 65.1%

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

   3. *-commutative 65.1%

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

   4. *-commutative 65.1%

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

   5. swap-sqr 65.1%

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

   6. swap-sqr 65.2%

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

   7. add-sqr-sqrt 65.2%

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

   8. pow2 65.2%

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

   9. metadata-eval 65.2%

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

6. Applied egg-rr 65.2%

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

   1. associate-*l* 65.2%

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

8. Simplified 65.2%

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

9. Taylor expanded in u2 around -inf 4.7%

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

10. Final simplification 4.7%

    \[\leadsto \left(u2 \cdot \sqrt{u1}\right) \cdot -6.28318530718 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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))))