Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 11.3s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% 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 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 98.3%

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

   3. *-commutative 98.3%

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

   4. *-commutative 98.3%

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

   5. swap-sqr 98.1%

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

   6. pow2 98.1%

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

   7. metadata-eval 98.6%

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

4. Applied egg-rr 98.6%

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

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0012000000569969416)
   (sqrt (* 39.47841760436263 (/ (* u2 u2) (+ (/ 1.0 u1) -1.0))))
   (* (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) <= 0.0012000000569969416f) {
		tmp = sqrtf((39.47841760436263f * ((u2 * u2) / ((1.0f / u1) + -1.0f))));
	} 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) <= 0.0012000000569969416e0) then
        tmp = sqrt((39.47841760436263e0 * ((u2 * u2) / ((1.0e0 / u1) + (-1.0e0)))))
    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(0.0012000000569969416))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(Float32(u2 * u2) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	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(0.0012000000569969416))
		tmp = sqrt((single(39.47841760436263) * ((u2 * u2) / ((single(1.0) / u1) + single(-1.0)))));
	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) < 0.00120000006

   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 98.3%

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

      1. sqrt-div 98.0%

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

      2. associate-*l/ 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. add-sqr-sqrt 97.6%

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

      2. sqrt-unprod 98.0%

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

      3. *-commutative 98.0%

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

      4. *-commutative 98.0%

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

      5. swap-sqr 97.8%

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

   7. Applied egg-rr 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-*l/ 98.9%

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

      3. *-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. metadata-eval 99.0%

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

   9. Simplified 99.0%

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

       1. unpow2 99.0%

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

   11. Applied egg-rr 99.0%

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

   if 0.00120000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 87.1%

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

      1. +-commutative 87.1%

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

   5. Simplified 87.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009999999776482582)
   (sqrt (* 39.47841760436263 (/ (* u2 u2) (+ (/ 1.0 u1) -1.0))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009999999776482582f) {
		tmp = sqrtf((39.47841760436263f * ((u2 * u2) / ((1.0f / u1) + -1.0f))));
	} 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) <= 0.009999999776482582e0) then
        tmp = sqrt((39.47841760436263e0 * ((u2 * u2) / ((1.0e0 / u1) + (-1.0e0)))))
    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(0.009999999776482582))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(Float32(u2 * u2) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	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(0.009999999776482582))
		tmp = sqrt((single(39.47841760436263) * ((u2 * u2) / ((single(1.0) / u1) + single(-1.0)))));
	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) < 0.00999999978

   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 96.5%

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

      1. sqrt-div 96.2%

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

      2. associate-*l/ 96.2%

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

   5. Applied egg-rr 96.2%

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

      1. add-sqr-sqrt 95.9%

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

      2. sqrt-unprod 96.2%

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

      3. *-commutative 96.2%

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

      4. *-commutative 96.2%

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

      5. swap-sqr 96.0%

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

   7. Applied egg-rr 96.9%

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

      1. *-commutative 96.9%

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

      2. associate-*l/ 97.0%

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

      3. *-lft-identity 97.0%

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

      4. times-frac 97.0%

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

      5. metadata-eval 97.0%

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

   9. Simplified 97.0%

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

       1. unpow2 97.0%

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

   11. Applied egg-rr 97.0%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

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

   3. Taylor expanded in u1 around 0 77.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2 \cdot u2}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 98.3%

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

   1. clear-num 98.2%

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

   2. sqrt-div 98.2%

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

   3. metadata-eval 98.2%

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

4. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. div-sub 98.3%

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

   4. *-inverses 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u1 around 0 98.3%

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

   1. neg-mul-1 98.3%

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

   2. sub-neg 98.3%

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

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. clear-num 98.2%

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

   2. sqrt-div 98.2%

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

   3. metadata-eval 98.2%

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

4. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. div-sub 98.3%

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

   4. *-inverses 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

   \[\leadsto \frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \]
8. 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 98.3%

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

3. Final simplification 98.3%

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

Alternative 7: 81.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 83.7%

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

   1. sqrt-div 83.4%

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

   2. associate-*l/ 83.4%

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

5. Applied egg-rr 83.4%

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

   1. add-sqr-sqrt 83.2%

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

   2. sqrt-unprod 83.4%

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

   3. *-commutative 83.4%

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

   4. *-commutative 83.4%

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

   5. swap-sqr 83.3%

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

7. Applied egg-rr 84.0%

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

   1. *-commutative 84.0%

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

   2. associate-*l/ 84.0%

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

   3. *-lft-identity 84.0%

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

   4. times-frac 84.1%

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

   5. metadata-eval 84.1%

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

9. Simplified 84.1%

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

    1. unpow2 84.1%

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

11. Applied egg-rr 84.1%

    \[\leadsto \sqrt{39.47841760436263 \cdot \frac{\color{blue}{u2 \cdot u2}}{\frac{1}{u1} + -1}} \]
12. Add Preprocessing

Alternative 8: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 83.7%

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

   1. sqrt-div 83.4%

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

   2. associate-*l/ 83.4%

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

5. Applied egg-rr 83.4%

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

   1. clear-num 83.5%

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

   2. un-div-inv 83.6%

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

   3. *-un-lft-identity 83.6%

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

   4. *-commutative 83.6%

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

   5. times-frac 83.5%

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

   6. sqrt-div 83.7%

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

   7. div-sub 83.7%

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

   8. *-inverses 83.7%

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

   9. sub-neg 83.7%

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

   10. metadata-eval 83.7%

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

7. Applied egg-rr 83.7%

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

   1. associate-*l/ 83.7%

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

   2. *-lft-identity 83.7%

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

9. Simplified 83.7%

   \[\leadsto \color{blue}{\frac{6.28318530718}{\frac{\sqrt{\frac{1}{u1} + -1}}{u2}}} \]
10. Add Preprocessing

Alternative 9: 81.6% 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 98.3%

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

3. Taylor expanded in u2 around 0 83.7%

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

4. Final simplification 83.7%

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

Alternative 10: 81.6% 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 98.3%

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

3. Taylor expanded in u2 around 0 83.7%

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

Alternative 11: 72.8% 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 98.3%

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

3. Taylor expanded in u2 around 0 83.7%

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

4. Taylor expanded in u1 around 0 73.7%

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

   1. +-commutative 85.4%

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

6. Simplified 73.7%

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

7. Final simplification 73.7%

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

Alternative 12: 64.3% accurate, 2.0× speedup

Math

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

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(u2 * Float32(sqrt(u1) * Float32(-Float32(-6.28318530718))))
end

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 83.7%

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

4. Taylor expanded in u1 around 0 65.6%

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

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. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 65.6%

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

7. Simplified 65.6%

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

8. Taylor expanded in u2 around 0 65.6%

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

   1. mul-1-neg 65.6%

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

10. Simplified 65.6%

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

11. Final simplification 65.6%

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

Alternative 13: 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 98.3%

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

3. Taylor expanded in u2 around 0 83.7%

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

4. Taylor expanded in u1 around 0 65.6%

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

5. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2024144 
(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))))