Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 9.7s

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

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

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. Final simplification 98.4%

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

Alternative 2: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.010999999940395355)
   (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) <= 0.010999999940395355f) {
		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) <= 0.010999999940395355e0) 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(0.010999999940395355))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u1 * Float32(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(0.010999999940395355))
		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) < 0.0109999999

   1. Initial program 98.7%

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

   2. Taylor expanded in u2 around 0 96.2%

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

      1. add-sqr-sqrt 95.8%

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

      2. sqrt-unprod 96.2%

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

      3. swap-sqr 96.0%

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

      4. metadata-eval 96.4%

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

      5. *-commutative 96.4%

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

      6. *-commutative 96.4%

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

      7. swap-sqr 96.5%

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

      8. add-sqr-sqrt 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in u2 around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. associate-*r/ 96.7%

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

      3. unpow2 96.7%

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

   7. Simplified 96.7%

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

   if 0.0109999999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.6%

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

   2. Taylor expanded in u1 around 0 71.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 81.0% accurate, 1.9× speedup

Math

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

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(u1 * Float32(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. Taylor expanded in u2 around 0 83.1%

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

   1. add-sqr-sqrt 82.9%

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

   2. sqrt-unprod 83.1%

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

   3. swap-sqr 83.0%

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

   4. metadata-eval 83.4%

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

   5. *-commutative 83.4%

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

   6. *-commutative 83.4%

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

   7. swap-sqr 83.4%

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

   8. add-sqr-sqrt 83.5%

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

4. Applied egg-rr 83.5%

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

5. Taylor expanded in u2 around 0 83.6%

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

   1. *-commutative 83.6%

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

   2. associate-*r/ 83.6%

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

   3. unpow2 83.6%

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

7. Simplified 83.6%

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

8. Final simplification 83.6%

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

Alternative 4: 80.6% 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. Taylor expanded in u2 around 0 83.1%

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

3. Final simplification 83.1%

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

Alternative 5: 80.9% 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 98.4%

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

2. Taylor expanded in u2 around 0 83.1%

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

   1. add-sqr-sqrt 82.9%

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

   2. sqrt-unprod 83.1%

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

   3. swap-sqr 83.0%

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

   4. metadata-eval 83.4%

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

   5. *-commutative 83.4%

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

   6. *-commutative 83.4%

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

   7. swap-sqr 83.4%

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

   8. add-sqr-sqrt 83.5%

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

4. Applied egg-rr 83.5%

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

   1. associate-*r* 83.5%

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

   2. sqrt-prod 83.4%

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

   3. sqrt-prod 82.9%

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

   4. add-sqr-sqrt 83.4%

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

6. Applied egg-rr 83.4%

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

   1. unpow1/2 83.4%

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

   2. *-commutative 83.4%

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

   3. unpow1/2 83.4%

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

   4. associate-*r/ 83.3%

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

8. Simplified 83.3%

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

9. Final simplification 83.3%

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

Alternative 6: 80.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * sqrt(((u1 / (single(1.0) - 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. Taylor expanded in u2 around 0 83.1%

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

   1. add-sqr-sqrt 82.9%

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

   2. sqrt-unprod 83.1%

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

   3. swap-sqr 83.0%

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

   4. metadata-eval 83.4%

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

   5. *-commutative 83.4%

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

   6. *-commutative 83.4%

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

   7. swap-sqr 83.4%

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

   8. add-sqr-sqrt 83.5%

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

4. Applied egg-rr 83.5%

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

   1. associate-*r* 83.5%

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

   2. sqrt-prod 83.4%

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

   3. sqrt-prod 82.9%

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

   4. add-sqr-sqrt 83.4%

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

6. Applied egg-rr 83.4%

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

7. Final simplification 83.4%

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

Alternative 7: 64.2% 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 83.1%

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

3. Taylor expanded in u1 around 0 65.4%

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

4. Final simplification 65.4%

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

Alternative 8: 19.3% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u1 around 0 86.7%

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

   1. unpow2 86.7%

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

   2. fma-udef 86.7%

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

4. Simplified 86.7%

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

5. Taylor expanded in u1 around inf 19.8%

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

6. Taylor expanded in u2 around 0 19.6%

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

7. Final simplification 19.6%

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

Reproduce

herbie shell --seed 2023229 
(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))))