Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 10.2s

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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. clear-num 98.1%

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

   2. sqrt-div 98.1%

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

   3. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

   1. div-sub 98.2%

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

   2. sub-neg 98.2%

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

   3. *-inverses 98.2%

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

   4. metadata-eval 98.2%

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

   5. +-commutative 98.2%

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

   6. remove-double-neg 98.2%

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

   7. unsub-neg 98.2%

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

   8. distribute-neg-frac 98.2%

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

   9. metadata-eval 98.2%

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

6. Simplified 98.2%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. sub-neg 98.4%

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

   4. neg-mul-1 98.4%

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

   5. frac-2neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. un-div-inv 98.4%

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

   8. metadata-eval 98.4%

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

   9. frac-2neg 98.4%

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

8. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

10. Simplified 98.4%

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

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00419999985024333:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.00419999985024333)
   (* u2 (* 6.28318530718 (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0)))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.00419999985024333f) {
		tmp = u2 * (6.28318530718f * sqrtf((1.0f / ((1.0f / u1) + -1.0f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (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 ((6.28318530718e0 * u2) <= 0.00419999985024333e0) then
        tmp = u2 * (6.28318530718e0 * sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0)))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + 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(0.00419999985024333))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.00419999985024333))
		tmp = u2 * (single(6.28318530718) * sqrt((single(1.0) / ((single(1.0) / u1) + single(-1.0)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (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) < 0.00419999985

   1. Initial program 98.4%

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

      1. clear-num 98.4%

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

      2. sqrt-div 98.3%

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

      3. metadata-eval 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. div-sub 98.3%

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

      2. sub-neg 98.3%

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

      3. *-inverses 98.3%

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

      4. metadata-eval 98.3%

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

      5. +-commutative 98.3%

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

      6. remove-double-neg 98.3%

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

      7. unsub-neg 98.3%

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

      8. distribute-neg-frac 98.3%

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

      9. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

      3. sub-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. frac-2neg 98.6%

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

      6. metadata-eval 98.6%

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

      7. un-div-inv 98.6%

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

      8. metadata-eval 98.6%

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

      9. frac-2neg 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in u2 around 0 97.5%

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

       1. associate-*r* 97.6%

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

       2. *-commutative 97.6%

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

       3. associate-*r* 97.8%

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

       4. sub-neg 97.8%

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

       5. metadata-eval 97.8%

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

   13. Simplified 97.8%

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

   if 0.00419999985 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0 89.5%

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

      1. +-commutative 89.5%

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

   5. Simplified 89.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.00419999985024333:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005200000014156103:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005200000014156103)
   (* u2 (* 6.28318530718 (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0)))))
   (/ (sin (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.005200000014156103f) {
		tmp = u2 * (6.28318530718f * sqrtf((1.0f / ((1.0f / u1) + -1.0f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) / 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 ((6.28318530718e0 * u2) <= 0.005200000014156103e0) then
        tmp = u2 * (6.28318530718e0 * sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0)))))
    else
        tmp = sin((6.28318530718e0 * u2)) / sqrt((1.0e0 / 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(0.005200000014156103))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.005200000014156103))
		tmp = u2 * (single(6.28318530718) * sqrt((single(1.0) / ((single(1.0) / u1) + single(-1.0)))));
	else
		tmp = sin((single(6.28318530718) * u2)) / 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) < 0.00520000001

   1. Initial program 98.4%

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

      1. clear-num 98.4%

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

      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. div-sub 98.3%

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

      2. sub-neg 98.3%

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

      3. *-inverses 98.3%

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

      4. metadata-eval 98.3%

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

      5. +-commutative 98.3%

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

      6. remove-double-neg 98.3%

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

      7. unsub-neg 98.3%

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

      8. distribute-neg-frac 98.3%

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

      9. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

      3. sub-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. frac-2neg 98.6%

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

      6. metadata-eval 98.6%

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

      7. un-div-inv 98.6%

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

      8. metadata-eval 98.6%

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

      9. frac-2neg 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in u2 around 0 97.4%

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

       1. associate-*r* 97.4%

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

       2. *-commutative 97.4%

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

       3. associate-*r* 97.6%

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

       4. sub-neg 97.6%

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

       5. metadata-eval 97.6%

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

   13. Simplified 97.6%

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

   if 0.00520000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

      1. clear-num 97.5%

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

      2. sqrt-div 97.8%

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

      3. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. div-sub 97.8%

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

      2. sub-neg 97.8%

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

      3. *-inverses 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

      6. remove-double-neg 97.8%

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

      7. unsub-neg 97.8%

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

      8. distribute-neg-frac 97.8%

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

      9. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

      1. associate-*l/ 98.1%

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

      2. *-un-lft-identity 98.1%

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

      3. sub-neg 98.1%

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

      4. neg-mul-1 98.1%

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

      5. frac-2neg 98.1%

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

      6. metadata-eval 98.1%

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

      7. un-div-inv 98.1%

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

      8. metadata-eval 98.1%

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

      9. frac-2neg 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. +-commutative 98.1%

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

   10. Simplified 98.1%

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

   11. Taylor expanded in u1 around 0 77.9%

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

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005200000014156103:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005200000014156103)
   (* u2 (* 6.28318530718 (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.005200000014156103f) {
		tmp = u2 * (6.28318530718f * sqrtf((1.0f / ((1.0f / u1) + -1.0f))));
	} 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) <= 0.005200000014156103e0) then
        tmp = u2 * (6.28318530718e0 * sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0)))))
    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(0.005200000014156103))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))));
	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(0.005200000014156103))
		tmp = u2 * (single(6.28318530718) * sqrt((single(1.0) / ((single(1.0) / u1) + single(-1.0)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00520000001

   1. Initial program 98.4%

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

      1. clear-num 98.4%

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

      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. div-sub 98.3%

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

      2. sub-neg 98.3%

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

      3. *-inverses 98.3%

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

      4. metadata-eval 98.3%

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

      5. +-commutative 98.3%

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

      6. remove-double-neg 98.3%

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

      7. unsub-neg 98.3%

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

      8. distribute-neg-frac 98.3%

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

      9. metadata-eval 98.3%

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

   6. Simplified 98.3%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

      3. sub-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. frac-2neg 98.6%

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

      6. metadata-eval 98.6%

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

      7. un-div-inv 98.6%

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

      8. metadata-eval 98.6%

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

      9. frac-2neg 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in u2 around 0 97.4%

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

       1. associate-*r* 97.4%

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

       2. *-commutative 97.4%

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

       3. associate-*r* 97.6%

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

       4. sub-neg 97.6%

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

       5. metadata-eval 97.6%

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

   13. Simplified 97.6%

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

   if 0.00520000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0 77.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005200000014156103:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 6: 87.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.014000000432133675)
   (*
    (sqrt (* u1 (+ 1.0 (* u1 (+ 1.0 u1)))))
    (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (/ (* 6.28318530718 u2) (sqrt (+ (/ 1.0 u1) -1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.014000000432133675f) {
		tmp = sqrtf((u1 * (1.0f + (u1 * (1.0f + u1))))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = (6.28318530718f * u2) / sqrtf(((1.0f / 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 ((u1 / (1.0e0 - u1)) <= 0.014000000432133675e0) then
        tmp = sqrt((u1 * (1.0e0 + (u1 * (1.0e0 + u1))))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = (6.28318530718e0 * u2) / sqrt(((1.0e0 / u1) + (-1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.014000000432133675))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(1.0) + u1))))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u1 / (single(1.0) - u1)) <= single(0.014000000432133675))
		tmp = sqrt((u1 * (single(1.0) + (u1 * (single(1.0) + u1))))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = (single(6.28318530718) * u2) / sqrt(((single(1.0) / u1) + single(-1.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.0140000004

   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 97.9%

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

   4. Taylor expanded in u2 around 0 87.9%

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

      1. unpow2 87.9%

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

   6. Applied egg-rr 87.9%

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

   if 0.0140000004 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   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. clear-num 98.0%

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

      2. sqrt-div 98.3%

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

      3. metadata-eval 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. div-sub 98.5%

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

      2. sub-neg 98.5%

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

      3. *-inverses 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. remove-double-neg 98.5%

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

      7. unsub-neg 98.5%

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

      8. distribute-neg-frac 98.5%

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

      9. metadata-eval 98.5%

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

   6. Simplified 98.5%

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

      1. associate-*l/ 98.6%

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

      2. *-un-lft-identity 98.6%

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

      3. sub-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. frac-2neg 98.6%

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

      6. metadata-eval 98.6%

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

      7. un-div-inv 98.6%

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

      8. metadata-eval 98.6%

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

      9. frac-2neg 98.6%

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

   8. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in u2 around 0 85.3%

       \[\leadsto \frac{\color{blue}{6.28318530718 \cdot u2}}{\sqrt{\frac{1}{u1} + -1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. clear-num 98.1%

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

   2. sqrt-div 98.1%

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

   3. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

   1. div-sub 98.2%

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

   2. sub-neg 98.2%

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

   3. *-inverses 98.2%

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

   4. metadata-eval 98.2%

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

   5. +-commutative 98.2%

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

   6. remove-double-neg 98.2%

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

   7. unsub-neg 98.2%

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

   8. distribute-neg-frac 98.2%

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

   9. metadata-eval 98.2%

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

6. Simplified 98.2%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. sub-neg 98.4%

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

   4. neg-mul-1 98.4%

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

   5. frac-2neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. un-div-inv 98.4%

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

   8. metadata-eval 98.4%

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

   9. frac-2neg 98.4%

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

8. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

10. Simplified 98.4%

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

11. Taylor expanded in u2 around 0 81.0%

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

    1. associate-*r* 81.0%

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

    2. *-commutative 81.0%

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

    3. associate-*r* 81.1%

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

    4. sub-neg 81.1%

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

    5. metadata-eval 81.1%

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

13. Simplified 81.1%

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

Alternative 8: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. clear-num 98.1%

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

   2. sqrt-div 98.1%

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

   3. metadata-eval 98.1%

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

4. Applied egg-rr 98.1%

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

   1. div-sub 98.2%

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

   2. sub-neg 98.2%

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

   3. *-inverses 98.2%

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

   4. metadata-eval 98.2%

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

   5. +-commutative 98.2%

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

   6. remove-double-neg 98.2%

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

   7. unsub-neg 98.2%

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

   8. distribute-neg-frac 98.2%

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

   9. metadata-eval 98.2%

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

6. Simplified 98.2%

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

   1. associate-*l/ 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. sub-neg 98.4%

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

   4. neg-mul-1 98.4%

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

   5. frac-2neg 98.4%

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

   6. metadata-eval 98.4%

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

   7. un-div-inv 98.4%

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

   8. metadata-eval 98.4%

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

   9. frac-2neg 98.4%

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

8. Applied egg-rr 98.4%

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

   1. +-commutative 98.4%

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

10. Simplified 98.4%

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

11. Taylor expanded in u2 around 0 81.1%

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

Alternative 9: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0 81.0%

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

4. Final simplification 81.0%

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

Alternative 10: 73.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0 81.0%

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

4. Taylor expanded in u1 around 0 73.6%

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

   1. +-commutative 88.1%

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

6. Simplified 73.6%

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

7. Final simplification 73.6%

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

Alternative 11: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 u2 around 0 81.0%

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

   1. sqrt-div 80.8%

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

   2. associate-*l/ 80.9%

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

5. Applied egg-rr 80.9%

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

6. Taylor expanded in u1 around 0 64.6%

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

   1. associate-*r* 64.7%

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

8. Simplified 64.7%

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

9. Final simplification 64.7%

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

Alternative 12: 64.7% accurate, 2.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0 81.0%

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

   1. sqrt-div 80.8%

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

   2. associate-*l/ 80.9%

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

5. Applied egg-rr 80.9%

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

6. Taylor expanded in u1 around 0 64.6%

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

   1. associate-*r* 64.7%

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

8. Simplified 64.7%

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

9. Taylor expanded in u1 around 0 64.6%

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

    1. *-commutative 64.6%

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

    2. associate-*r* 64.7%

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

11. Simplified 64.7%

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

Alternative 13: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0 81.0%

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

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 2024123 
(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))))