Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 12.0s

Alternatives: 12

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: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   7. metadata-eval 98.5%

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

4. Applied egg-rr 98.5%

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

Alternative 2: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.000859999970998615:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(\frac{u1}{1 - u1} \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.000859999970998615)
   (sqrt (* 39.47841760436263 (* u2 (* (/ u1 (- 1.0 u1)) u2))))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.000859999970998615f) {
		tmp = sqrtf((39.47841760436263f * (u2 * ((u1 / (1.0f - u1)) * u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.000859999970998615e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 * ((u1 / (1.0e0 - u1)) * u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.000859999970998615))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 * Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.000859999970998615))
		tmp = sqrt((single(39.47841760436263) * (u2 * ((u1 / (single(1.0) - u1)) * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 8.59999971e-4

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 98.3%

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

      1. clear-num 98.4%

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

      2. sqrt-div 98.2%

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

      3. metadata-eval 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. add-sqr-sqrt 97.5%

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

      2. sqrt-unprod 98.2%

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

      3. *-commutative 98.2%

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

      4. *-commutative 98.2%

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

      5. swap-sqr 98.1%

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

   7. Applied egg-rr 98.9%

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

      1. div-inv 98.8%

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

      2. unpow2 98.8%

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

      3. associate-*l* 98.8%

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

      4. metadata-eval 98.8%

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

      5. sub-neg 98.8%

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

      6. *-inverses 98.8%

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

      7. div-sub 98.9%

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

      8. clear-num 99.0%

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

   9. Applied egg-rr 99.0%

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

   if 8.59999971e-4 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0 89.7%

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

      1. +-commutative 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.000859999970998615:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(\frac{u1}{1 - u1} \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(\frac{u1}{1 - u1} \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.019999999552965164)
   (sqrt (* 39.47841760436263 (* u2 (* (/ u1 (- 1.0 u1)) u2))))
   (/ (sin (* u2 6.28318530718)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.019999999552965164f) {
		tmp = sqrtf((39.47841760436263f * (u2 * ((u1 / (1.0f - u1)) * u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) / sqrtf((1.0f / u1));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.019999999552965164e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 * ((u1 / (1.0e0 - u1)) * u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) / sqrt((1.0e0 / u1))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.019999999552965164))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 * Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.019999999552965164))
		tmp = sqrt((single(39.47841760436263) * (u2 * ((u1 / (single(1.0) - u1)) * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0199999996

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 95.5%

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

      1. clear-num 95.5%

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

      2. sqrt-div 95.4%

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

      3. metadata-eval 95.4%

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

   5. Applied egg-rr 95.4%

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

      1. add-sqr-sqrt 94.8%

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

      2. sqrt-unprod 95.4%

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

      3. *-commutative 95.4%

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

      4. *-commutative 95.4%

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

      5. swap-sqr 95.3%

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

   7. Applied egg-rr 95.9%

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

      1. div-inv 95.9%

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

      2. unpow2 95.9%

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

      3. associate-*l* 95.9%

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

      4. metadata-eval 95.9%

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

      5. sub-neg 95.9%

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

      6. *-inverses 95.9%

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

      7. div-sub 95.9%

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

      8. clear-num 96.0%

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

   9. Applied egg-rr 96.0%

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

   if 0.0199999996 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

      1. add-sqr-sqrt 97.0%

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

      2. sqrt-unprod 98.1%

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

      3. *-commutative 98.1%

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

      4. *-commutative 98.1%

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

      5. swap-sqr 97.6%

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

      6. pow2 97.6%

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

      7. metadata-eval 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. *-commutative 98.0%

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

      2. *-commutative 98.0%

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

      3. sqrt-prod 97.4%

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

      4. metadata-eval 98.1%

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

      5. sqrt-pow1 98.1%

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

      6. metadata-eval 98.1%

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

      7. pow1 98.1%

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

      8. sqrt-div 97.8%

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

      9. clear-num 97.5%

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

      10. div-inv 97.9%

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

      11. *-commutative 97.9%

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

      12. sqrt-undiv 98.3%

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

      13. div-sub 98.2%

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

      14. *-inverses 98.2%

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

      15. sub-neg 98.2%

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

      16. metadata-eval 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in u1 around 0 80.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(\frac{u1}{1 - u1} \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(\frac{u1}{1 - u1} \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.019999999552965164)
   (sqrt (* 39.47841760436263 (* u2 (* (/ u1 (- 1.0 u1)) u2))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.019999999552965164f) {
		tmp = sqrtf((39.47841760436263f * (u2 * ((u1 / (1.0f - u1)) * u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.019999999552965164e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 * ((u1 / (1.0e0 - u1)) * u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.019999999552965164))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 * Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.019999999552965164))
		tmp = sqrt((single(39.47841760436263) * (u2 * ((u1 / (single(1.0) - u1)) * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0199999996

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 95.5%

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

      1. clear-num 95.5%

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

      2. sqrt-div 95.4%

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

      3. metadata-eval 95.4%

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

   5. Applied egg-rr 95.4%

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

      1. add-sqr-sqrt 94.8%

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

      2. sqrt-unprod 95.4%

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

      3. *-commutative 95.4%

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

      4. *-commutative 95.4%

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

      5. swap-sqr 95.3%

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

   7. Applied egg-rr 95.9%

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

      1. div-inv 95.9%

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

      2. unpow2 95.9%

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

      3. associate-*l* 95.9%

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

      4. metadata-eval 95.9%

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

      5. sub-neg 95.9%

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

      6. *-inverses 95.9%

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

      7. div-sub 95.9%

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

      8. clear-num 96.0%

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

   9. Applied egg-rr 96.0%

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

   if 0.0199999996 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 79.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(u2 \cdot \left(\frac{u1}{1 - u1} \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.6%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   7. metadata-eval 98.5%

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

4. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. sqrt-prod 97.9%

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

   4. metadata-eval 98.5%

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

   5. sqrt-pow1 98.5%

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

   6. metadata-eval 98.5%

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

   7. pow1 98.5%

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

   8. sqrt-div 98.2%

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

   9. clear-num 98.1%

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

   10. div-inv 98.2%

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

   11. *-commutative 98.2%

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

   12. sqrt-undiv 98.5%

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

   13. div-sub 98.4%

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

   14. *-inverses 98.4%

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

   15. sub-neg 98.4%

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

   16. metadata-eval 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in u1 around 0 98.5%

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

   1. neg-mul-1 98.5%

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

   2. sub-neg 98.5%

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

9. Simplified 98.5%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Final simplification 98.5%

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

Alternative 7: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 80.9%

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

   1. clear-num 80.9%

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

   2. sqrt-div 80.8%

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

   3. metadata-eval 80.8%

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

5. Applied egg-rr 80.8%

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

   1. add-sqr-sqrt 80.4%

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

   2. sqrt-unprod 80.8%

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

   3. *-commutative 80.8%

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

   4. *-commutative 80.8%

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

   5. swap-sqr 80.7%

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

7. Applied egg-rr 81.2%

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

   1. div-inv 81.1%

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

   2. unpow2 81.1%

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

   3. associate-*l* 81.1%

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

   4. metadata-eval 81.1%

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

   5. sub-neg 81.1%

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

   6. *-inverses 81.1%

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

   7. div-sub 81.2%

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

   8. clear-num 81.2%

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

9. Applied egg-rr 81.2%

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

10. Final simplification 81.2%

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

Alternative 8: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 80.9%

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

4. Final simplification 80.9%

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

Alternative 9: 81.1% 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 98.5%

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

3. Taylor expanded in u2 around 0 80.9%

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

Alternative 10: 72.5% 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 98.5%

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

3. Taylor expanded in u2 around 0 80.9%

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

4. Taylor expanded in u1 around 0 74.0%

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

   1. +-commutative 88.3%

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

6. Simplified 74.0%

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

7. Final simplification 74.0%

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

Alternative 11: 64.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * (1.0f / powf(u1, -0.5f)));
}

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 * (1.0e0 / (u1 ** (-0.5e0))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * (single(1.0) / (u1 ^ single(-0.5))));
end

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 80.9%

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

4. Taylor expanded in u1 around 0 64.6%

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

   1. pow1/2 64.6%

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

   2. metadata-eval 64.6%

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

   3. pow-flip 64.7%

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

6. Applied egg-rr 64.7%

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

7. Final simplification 64.7%

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

Alternative 12: 64.3% 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 98.5%

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

3. Taylor expanded in u2 around 0 80.9%

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

4. Taylor expanded in u1 around 0 64.6%

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

5. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2024146 
(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))))