Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 9.6s

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. clear-num 98.5%

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

   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. associate-*l/ 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. clear-num 98.3%

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

   4. div-sub 98.3%

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

   5. *-inverses 98.3%

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

   6. sub-neg 98.3%

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

   7. metadata-eval 98.3%

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

6. Applied egg-rr 98.3%

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

   1. associate-/r/ 98.3%

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

   2. metadata-eval 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. distribute-lft-neg-out 98.3%

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

   5. /-rgt-identity 98.3%

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

   6. times-frac 98.6%

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

   7. *-commutative 98.6%

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

   8. *-rgt-identity 98.6%

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

   9. distribute-frac-neg2 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-/l* 98.6%

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

   12. neg-mul-1 98.6%

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

   13. distribute-frac-neg2 98.6%

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

   14. remove-double-neg 98.6%

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

8. Simplified 98.6%

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

   1. metadata-eval 98.6%

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

   2. sub-neg 98.6%

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

   3. *-inverses 98.6%

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

   4. div-sub 98.7%

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

   5. clear-num 98.7%

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

10. Applied egg-rr 98.7%

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

Alternative 2: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0020000000949949026:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.0020000000949949026)
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))
   (/ (* 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)) <= 0.0020000000949949026f) {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (1.0f + u1)));
	} 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)) <= 0.0020000000949949026e0) then
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + u1)))
    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(0.0020000000949949026))
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	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(0.0020000000949949026))
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (single(1.0) + u1)));
	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)) < 0.00200000009

   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 u1 around 0 97.9%

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

      1. +-commutative 97.9%

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

   5. Simplified 97.9%

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

   if 0.00200000009 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.7%

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

      1. clear-num 98.7%

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

      2. sqrt-div 98.6%

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

      3. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in u2 around 0 81.5%

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

      1. associate-*l/ 81.7%

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

      2. *-un-lft-identity 81.7%

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

      3. div-sub 81.6%

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

      4. *-inverses 81.6%

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

      5. sub-neg 81.6%

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

      6. metadata-eval 81.6%

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

   7. Applied egg-rr 81.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0020000000949949026:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.01600000075995922:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.01600000075995922)
   (/ (* 6.28318530718 u2) (sqrt (+ (/ 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) <= 0.01600000075995922f) {
		tmp = (6.28318530718f * u2) / sqrtf(((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) <= 0.01600000075995922e0) then
        tmp = (6.28318530718e0 * u2) / sqrt(((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(0.01600000075995922))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / sqrt(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(0.01600000075995922))
		tmp = (single(6.28318530718) * u2) / sqrt(((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) < 0.0160000008

   1. Initial program 98.7%

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

      1. clear-num 98.6%

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

      2. sqrt-div 98.4%

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

      3. metadata-eval 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in u2 around 0 95.3%

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

      1. associate-*l/ 95.7%

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

      2. *-un-lft-identity 95.7%

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

      3. div-sub 95.6%

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

      4. *-inverses 95.6%

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

      5. sub-neg 95.6%

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

      6. metadata-eval 95.6%

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

   7. Applied egg-rr 95.6%

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

   if 0.0160000008 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   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 98.3%

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

      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. 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. clear-num 97.8%

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

      4. div-sub 97.8%

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

      5. *-inverses 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. associate-/r/ 98.2%

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

      2. metadata-eval 98.2%

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

      3. distribute-neg-frac 98.2%

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

      4. distribute-lft-neg-out 98.2%

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

      5. /-rgt-identity 98.2%

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

      6. times-frac 98.2%

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

      7. *-commutative 98.2%

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

      8. *-rgt-identity 98.2%

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

      9. distribute-frac-neg2 98.2%

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

      10. *-commutative 98.2%

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

      11. associate-/l* 98.2%

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

      12. neg-mul-1 98.2%

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

      13. distribute-frac-neg2 98.2%

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

      14. remove-double-neg 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in u1 around 0 71.2%

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

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.01600000075995922:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.01600000075995922)
   (/ (* 6.28318530718 u2) (sqrt (+ (/ 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) <= 0.01600000075995922f) {
		tmp = (6.28318530718f * u2) / sqrtf(((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) <= 0.01600000075995922e0) then
        tmp = (6.28318530718e0 * u2) / sqrt(((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(0.01600000075995922))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / sqrt(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(0.01600000075995922))
		tmp = (single(6.28318530718) * u2) / sqrt(((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) < 0.0160000008

   1. Initial program 98.7%

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

      1. clear-num 98.6%

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

      2. sqrt-div 98.4%

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

      3. metadata-eval 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in u2 around 0 95.3%

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

      1. associate-*l/ 95.7%

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

      2. *-un-lft-identity 95.7%

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

      3. div-sub 95.6%

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

      4. *-inverses 95.6%

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

      5. sub-neg 95.6%

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

      6. metadata-eval 95.6%

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

   7. Applied egg-rr 95.6%

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

   if 0.0160000008 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   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 u1 around 0 71.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.01600000075995922:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. clear-num 98.5%

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

   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. associate-*l/ 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. clear-num 98.3%

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

   4. div-sub 98.3%

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

   5. *-inverses 98.3%

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

   6. sub-neg 98.3%

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

   7. metadata-eval 98.3%

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

6. Applied egg-rr 98.3%

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

   1. associate-/r/ 98.3%

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

   2. metadata-eval 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. distribute-lft-neg-out 98.3%

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

   5. /-rgt-identity 98.3%

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

   6. times-frac 98.6%

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

   7. *-commutative 98.6%

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

   8. *-rgt-identity 98.6%

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

   9. distribute-frac-neg2 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-/l* 98.6%

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

   12. neg-mul-1 98.6%

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

   13. distribute-frac-neg2 98.6%

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

   14. remove-double-neg 98.6%

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

8. Simplified 98.6%

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

9. Taylor expanded in u1 around 0 98.7%

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

    1. neg-mul-1 98.7%

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

    2. sub-neg 98.7%

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

11. Simplified 98.7%

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

Alternative 6: 98.3% 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 98.6%

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

   1. clear-num 98.5%

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

   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. associate-*l/ 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. clear-num 98.3%

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

   4. div-sub 98.3%

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

   5. *-inverses 98.3%

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

   6. sub-neg 98.3%

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

   7. metadata-eval 98.3%

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

6. Applied egg-rr 98.3%

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

   1. associate-/r/ 98.3%

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

   2. metadata-eval 98.3%

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

   3. distribute-neg-frac 98.3%

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

   4. distribute-lft-neg-out 98.3%

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

   5. /-rgt-identity 98.3%

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

   6. times-frac 98.6%

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

   7. *-commutative 98.6%

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

   8. *-rgt-identity 98.6%

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

   9. distribute-frac-neg2 98.6%

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

   10. *-commutative 98.6%

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

   11. associate-/l* 98.6%

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

   12. neg-mul-1 98.6%

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

   13. distribute-frac-neg2 98.6%

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

   14. remove-double-neg 98.6%

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

8. Simplified 98.6%

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

Alternative 7: 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 98.6%

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

3. Final simplification 98.6%

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

Alternative 8: 81.6% 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 98.6%

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

   1. clear-num 98.5%

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

   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. Taylor expanded in u2 around 0 83.5%

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

   1. associate-*l/ 83.7%

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

   2. *-un-lft-identity 83.7%

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

   3. div-sub 83.7%

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

   4. *-inverses 83.7%

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

   5. sub-neg 83.7%

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

   6. metadata-eval 83.7%

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

7. Applied egg-rr 83.7%

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

Alternative 9: 81.7% accurate, 1.9× speedup

Math

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

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(Float32(6.28318530718) * 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 98.6%

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

3. Taylor expanded in u2 around 0 83.6%

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

4. Final simplification 83.6%

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

Alternative 10: 81.7% 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 98.6%

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

3. Taylor expanded in u2 around 0 83.5%

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

4. Final simplification 83.5%

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

Alternative 11: 73.1% 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 98.6%

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

3. Taylor expanded in u2 around 0 83.5%

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

4. Taylor expanded in u1 around 0 75.8%

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

   1. +-commutative 86.8%

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

6. Simplified 75.8%

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

7. Final simplification 75.8%

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

Alternative 12: 64.8% 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 98.6%

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

3. Taylor expanded in u2 around 0 83.5%

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

4. Taylor expanded in u1 around 0 66.0%

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

   1. *-commutative 66.0%

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

   2. associate-*l* 66.0%

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

   3. *-commutative 66.0%

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

6. Simplified 66.0%

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

7. Final simplification 66.0%

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

Alternative 13: 64.8% 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 98.6%

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

3. Taylor expanded in u2 around 0 83.5%

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

4. Taylor expanded in u1 around 0 66.0%

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

5. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2024137 
(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))))