Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 7.1s

Alternatives: 10

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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. clear-num N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. lower-/.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Final simplification 98.3%

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

Alternative 2: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.07500000298023224:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot t\_0 - -6.28318530718 \cdot t\_0\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* u2 6.28318530718) 0.07500000298023224)
     (*
      (- (* (* -41.341702240407926 (* u2 u2)) t_0) (* -6.28318530718 t_0))
      u2)
     (* (sqrt (* (- u1 -1.0) u1)) (sin (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.07500000298023224f) {
		tmp = (((-41.341702240407926f * (u2 * u2)) * t_0) - (-6.28318530718f * t_0)) * u2;
	} else {
		tmp = sqrtf(((u1 - -1.0f) * u1)) * sinf((u2 * 6.28318530718f));
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    if ((u2 * 6.28318530718e0) <= 0.07500000298023224e0) then
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) * t_0) - ((-6.28318530718e0) * t_0)) * u2
    else
        tmp = sqrt(((u1 - (-1.0e0)) * u1)) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.07500000298023224))
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) * t_0) - Float32(Float32(-6.28318530718) * t_0)) * u2);
	else
		tmp = Float32(sqrt(Float32(Float32(u1 - Float32(-1.0)) * u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.07500000298023224))
		tmp = (((single(-41.341702240407926) * (u2 * u2)) * t_0) - (single(-6.28318530718) * t_0)) * u2;
	else
		tmp = sqrt(((u1 - single(-1.0)) * u1)) * sin((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.075000003

   1. Initial program 98.5%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   5. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, -41.341702240407926, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.2%

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

   if 0.075000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. lower-+.f32 84.9

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

   5. Applied rewrites 84.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.07500000298023224:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} - -6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.20000000298023224:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot t\_0 - -6.28318530718 \cdot t\_0\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* u2 6.28318530718) 0.20000000298023224)
     (*
      (- (* (* -41.341702240407926 (* u2 u2)) t_0) (* -6.28318530718 t_0))
      u2)
     (* (sqrt u1) (sin (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.20000000298023224f) {
		tmp = (((-41.341702240407926f * (u2 * u2)) * t_0) - (-6.28318530718f * t_0)) * u2;
	} else {
		tmp = sqrtf(u1) * sinf((u2 * 6.28318530718f));
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    if ((u2 * 6.28318530718e0) <= 0.20000000298023224e0) then
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) * t_0) - ((-6.28318530718e0) * t_0)) * u2
    else
        tmp = sqrt(u1) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.20000000298023224))
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) * t_0) - Float32(Float32(-6.28318530718) * t_0)) * u2);
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.20000000298023224))
		tmp = (((single(-41.341702240407926) * (u2 * u2)) * t_0) - (single(-6.28318530718) * t_0)) * u2;
	else
		tmp = sqrt(u1) * sin((u2 * single(6.28318530718)));
	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 98.5%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, -41.341702240407926, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

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

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

   1. Initial program 97.1%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}} \]

      3. lower-sin.f32 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{u1} \]

      4. lower-*.f32 N/A

         \[\leadsto \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{u1} \]

      5. lower-sqrt.f32 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 93.5%

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

Alternative 5: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot t\_0 - -6.28318530718 \cdot t\_0\right) \cdot u2 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (- (* (* -41.341702240407926 (* u2 u2)) t_0) (* -6.28318530718 t_0))
    u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return (((-41.341702240407926f * (u2 * u2)) * t_0) - (-6.28318530718f * t_0)) * 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
    real(4) :: t_0
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    code = ((((-41.341702240407926e0) * (u2 * u2)) * t_0) - ((-6.28318530718e0) * t_0)) * u2
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) * t_0) - Float32(Float32(-6.28318530718) * t_0)) * u2)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = (((single(-41.341702240407926) * (u2 * u2)) * t_0) - (single(-6.28318530718) * t_0)) * u2;
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

5. Applied rewrites 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, -41.341702240407926, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]
6. Step-by-step derivation

7. Applied rewrites 87.1%

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

8. Final simplification 87.1%

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

Alternative 6: 89.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return ((((-41.341702240407926f * u2) * u2) - -6.28318530718f) * sqrtf((u1 / (1.0f - u1)))) * u2;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((((-41.341702240407926e0) * u2) * u2) - (-6.28318530718e0)) * sqrt((u1 / (1.0e0 - u1)))) * u2
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

5. Applied rewrites 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, -41.341702240407926, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]
6. Step-by-step derivation

7. Applied rewrites 87.1%

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

8. Final simplification 87.1%

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

Alternative 7: 89.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * sqrtf((u1 / (1.0f - u1)))) * u2;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * sqrt((u1 / (1.0e0 - u1)))) * u2
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

5. Applied rewrites 79.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}, -41.341702240407926, \sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2} \]
6. Step-by-step derivation

7. Applied rewrites 87.1%

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

8. Final simplification 87.1%

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

Alternative 8: 82.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{314159265359}{50000000000}\right)} \cdot u2 \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{314159265359}{50000000000}\right)} \cdot u2 \]

   5. lower-sqrt.f32 N/A

      \[\leadsto \left(\color{blue}{\sqrt{\frac{u1}{1 - u1}}} \cdot \frac{314159265359}{50000000000}\right) \cdot u2 \]

   6. lower-/.f32 N/A

      \[\leadsto \left(\sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \frac{314159265359}{50000000000}\right) \cdot u2 \]

   7. lower--.f32 79.4

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

5. Applied rewrites 79.4%

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

7. Applied rewrites 79.9%

   \[\leadsto \sqrt{39.47841760436263 \cdot \frac{u1}{1 - u1}} \cdot u2 \]
8. Add Preprocessing

Alternative 9: 73.3% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. lift-sqrt.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   5. clear-num N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

   6. sqrt-div N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

   7. metadata-eval N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

   8. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   9. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{\sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{\sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1 - u1}{u1}}} \]

   13. lower-sqrt.f32 N/A

       \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   14. lower-/.f32 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\color{blue}{\frac{1 - u1}{u1}}}} \]

   2. lift--.f32 N/A

      \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{\color{blue}{1 - u1}}{u1}}} \]

   3. div-sub N/A

      \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

   4. *-inverses N/A

      \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} - \color{blue}{1}}} \]

   5. lower--.f32 N/A

      \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\color{blue}{\frac{1}{u1} - 1}}} \]

   6. lower-/.f32 98.0

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

6. Applied rewrites 98.0%

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

7. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(u2 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

   4. lower-sqrt.f32 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} - 1}}} \]

   6. lower--.f32 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} - 1}}} \]

   7. lower-/.f32 79.6

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

9. Applied rewrites 79.6%

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

10. Taylor expanded in u1 around 0

    \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)} \]
11. Step-by-step derivation

12. Applied rewrites 72.5%

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

13. Final simplification 72.5%

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

Alternative 10: 65.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{314159265359}{50000000000}\right)} \cdot u2 \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{314159265359}{50000000000}\right)} \cdot u2 \]

   5. lower-sqrt.f32 N/A

      \[\leadsto \left(\color{blue}{\sqrt{\frac{u1}{1 - u1}}} \cdot \frac{314159265359}{50000000000}\right) \cdot u2 \]

   6. lower-/.f32 N/A

      \[\leadsto \left(\sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \frac{314159265359}{50000000000}\right) \cdot u2 \]

   7. lower--.f32 79.4

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

5. Applied rewrites 79.4%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(\sqrt{u1} \cdot u2\right)} \]
7. Step-by-step derivation

8. Applied rewrites 64.3%

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

10. Applied rewrites 64.3%

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

11. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2024312 
(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))))