Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 11.2s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. 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. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

Alternative 2: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.14000000059604645:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.14000000059604645)
   (*
    u2
    (*
     (sqrt (/ u1 (- 1.0 u1)))
     (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (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) <= 0.14000000059604645f) {
		tmp = u2 * (sqrtf((u1 / (1.0f - u1))) * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} 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) <= 0.14000000059604645e0) then
        tmp = u2 * (sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    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(0.14000000059604645))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	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(0.14000000059604645))
		tmp = u2 * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	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) < 0.140000001

   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 98.7%

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

      2. inv-pow 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. unpow-1 98.7%

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

      2. *-inverses 98.7%

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

      3. div-sub 98.6%

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

      4. *-inverses 98.6%

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

      5. associate-/r* 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. times-frac 98.6%

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

      8. sub-neg 98.6%

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

      9. metadata-eval 98.6%

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

      10. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in u2 around 0 98.1%

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

      1. associate-*r* 98.1%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

      2. distribute-rgt-out 98.0%

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

      3. sub-neg 98.0%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      4. *-inverses 98.0%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \left(-\color{blue}{\frac{-u1}{-u1}}\right)}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      5. distribute-frac-neg 98.0%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{\frac{-\left(-u1\right)}{-u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      6. remove-double-neg 98.0%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \frac{\color{blue}{u1}}{-u1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      7. distribute-frac-neg2 98.0%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{\left(-\frac{u1}{u1}\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      8. sub-neg 98.0%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      9. div-sub 98.0%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      10. associate-/r/ 97.9%

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\frac{1}{1 - u1} \cdot u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      11. associate-*l/ 98.0%

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\frac{1 \cdot u1}{1 - u1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      12. *-lft-identity 98.0%

          \[\leadsto u2 \cdot \left(\sqrt{\frac{\color{blue}{u1}}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   9. Simplified 98.0%

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

       1. unpow2 98.0%

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

   11. Applied egg-rr 98.0%

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

   if 0.140000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

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

   3. Taylor expanded in u1 around 0 88.4%

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

      1. +-commutative 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.14000000059604645:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf(((1.0f / u1) * (u1 / (-1.0f + (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) * (u1 / ((-1.0e0) + (1.0e0 / 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) * Float32(u1 / Float32(Float32(-1.0) + 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) * (u1 / (single(-1.0) + (single(1.0) / u1)))));
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. 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. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. unpow-1 98.5%

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

   2. *-inverses 98.5%

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

   3. div-sub 98.5%

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

   4. *-inverses 98.5%

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

   5. associate-/r* 98.4%

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

   6. *-un-lft-identity 98.4%

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

   7. times-frac 98.4%

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

   8. sub-neg 98.4%

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

   9. metadata-eval 98.4%

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

   10. +-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 4: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf(((1.0f / u1) * (u1 * (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) * (u1 * (u1 / (1.0e0 - 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) * Float32(u1 * 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) * (u1 * (u1 / (single(1.0) - u1)))));
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. 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. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. unpow-1 98.5%

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

   2. *-inverses 98.5%

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

   3. div-sub 98.5%

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

   4. *-inverses 98.5%

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

   5. associate-/r* 98.4%

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

   6. *-un-lft-identity 98.4%

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

   7. times-frac 98.4%

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

   8. sub-neg 98.4%

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

   9. metadata-eval 98.4%

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

   10. +-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

   1. clear-num 98.3%

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

   2. associate-/r/ 98.4%

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

   3. lft-mult-inverse 98.4%

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

   4. associate-*r/ 98.4%

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

   5. frac-times 98.4%

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

   6. *-un-lft-identity 98.4%

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

   7. +-commutative 98.4%

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

   8. distribute-rgt-in 98.3%

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

   9. lft-mult-inverse 98.4%

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

   10. neg-mul-1 98.4%

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

   11. /-rgt-identity 98.4%

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

   12. sub-neg 98.4%

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

   13. /-rgt-identity 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 5: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.20000000298023224:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.20000000298023224)
   (*
    u2
    (*
     (sqrt (/ u1 (- 1.0 u1)))
     (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.20000000298023224f) {
		tmp = u2 * (sqrtf((u1 / (1.0f - u1))) * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} 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) <= 0.20000000298023224e0) then
        tmp = u2 * (sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    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(0.20000000298023224))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	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(0.20000000298023224))
		tmp = u2 * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	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) < 0.200000003

   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 98.6%

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

      2. inv-pow 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. unpow-1 98.6%

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

      2. *-inverses 98.6%

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

      3. div-sub 98.6%

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

      4. *-inverses 98.6%

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

      5. associate-/r* 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. times-frac 98.6%

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

      8. sub-neg 98.6%

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

      9. metadata-eval 98.6%

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

      10. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in u2 around 0 97.6%

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

      1. associate-*r* 97.6%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

      2. distribute-rgt-out 97.5%

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

      3. sub-neg 97.5%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      4. *-inverses 97.5%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \left(-\color{blue}{\frac{-u1}{-u1}}\right)}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      5. distribute-frac-neg 97.5%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{\frac{-\left(-u1\right)}{-u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      6. remove-double-neg 97.5%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \frac{\color{blue}{u1}}{-u1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      7. distribute-frac-neg2 97.5%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{\left(-\frac{u1}{u1}\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      8. sub-neg 97.5%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      9. div-sub 97.6%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      10. associate-/r/ 97.4%

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\frac{1}{1 - u1} \cdot u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      11. associate-*l/ 97.5%

          \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\frac{1 \cdot u1}{1 - u1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      12. *-lft-identity 97.5%

          \[\leadsto u2 \cdot \left(\sqrt{\frac{\color{blue}{u1}}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   9. Simplified 97.5%

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

       1. unpow2 97.5%

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

   11. Applied egg-rr 97.5%

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

   if 0.200000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

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

   3. Taylor expanded in u1 around 0 80.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.20000000298023224:\\ \;\;\;\;u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 7: 88.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf((u1 / (1.0f - u1))) * (6.28318530718f + (-41.341702240407926f * (u2 * 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 = u2 * (sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
end

Derivation

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 98.5%

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

   2. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. unpow-1 98.5%

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

   2. *-inverses 98.5%

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

   3. div-sub 98.5%

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

   4. *-inverses 98.5%

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

   5. associate-/r* 98.4%

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

   6. *-un-lft-identity 98.4%

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

   7. times-frac 98.4%

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

   8. sub-neg 98.4%

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

   9. metadata-eval 98.4%

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

   10. +-commutative 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in u2 around 0 89.0%

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

   1. associate-*r* 89.0%

      \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

   2. distribute-rgt-out 88.9%

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

   3. sub-neg 88.9%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   4. *-inverses 88.9%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \left(-\color{blue}{\frac{-u1}{-u1}}\right)}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   5. distribute-frac-neg 88.9%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{\frac{-\left(-u1\right)}{-u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   6. remove-double-neg 88.9%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \frac{\color{blue}{u1}}{-u1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   7. distribute-frac-neg2 88.9%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{\left(-\frac{u1}{u1}\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   8. sub-neg 88.9%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   9. div-sub 88.9%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   10. associate-/r/ 88.8%

       \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\frac{1}{1 - u1} \cdot u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   11. associate-*l/ 88.9%

       \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\frac{1 \cdot u1}{1 - u1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   12. *-lft-identity 88.9%

       \[\leadsto u2 \cdot \left(\sqrt{\frac{\color{blue}{u1}}{1 - u1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

9. Simplified 88.9%

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

    1. unpow2 88.9%

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

11. Applied egg-rr 88.9%

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

12. Final simplification 88.9%

    \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \]
13. Add Preprocessing

Alternative 8: 80.9% 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 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

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

   1. *-commutative 80.6%

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

   2. associate-*r* 80.7%

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

5. Simplified 80.7%

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

Alternative 9: 80.9% 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 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

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

4. Final simplification 80.6%

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

Alternative 10: 72.6% 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 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

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

4. Taylor expanded in u1 around 0 72.6%

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

   1. +-commutative 87.2%

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

6. Simplified 72.6%

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

7. Final simplification 72.6%

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

Alternative 11: 64.4% 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 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

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

4. Taylor expanded in u1 around 0 64.9%

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

   1. *-commutative 64.9%

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

   2. associate-*l* 64.9%

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

   3. *-commutative 64.9%

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

6. Simplified 64.9%

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

7. Final simplification 64.9%

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

Alternative 12: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

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

4. Taylor expanded in u1 around 0 64.9%

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

5. Final simplification 64.9%

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

Reproduce

herbie shell --seed 2024191 
(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))))