Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 8.3s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Final simplification 98.2%

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

Alternative 2: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.02500000037252903:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.02500000037252903)
   (* (* u2 6.28318530718) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt u1) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.02500000037252903f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((u1 / (1.0f - u1)));
	} 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) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.02500000037252903e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(u1) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.02500000037252903))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.02500000037252903))
		tmp = (u2 * single(6.28318530718)) * sqrt((u1 / (single(1.0) - u1)));
	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) < 0.0250000004

   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

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

   5. Applied rewrites 95.6%

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

   if 0.0250000004 < (*.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. lower-sqrt.f32 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.02500000037252903:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 80.5%

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

6. Final simplification 80.5%

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

Alternative 4: 64.4% 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 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 80.5%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 62.6%

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

9. Final simplification 62.6%

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

Alternative 5: 64.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (sqrtf(u1) * 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) * 6.28318530718e0) * u2
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 80.5%

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

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 62.6%

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

10. Applied rewrites 62.6%

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

Alternative 6: 19.6% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ u2 \cdot 6.28318530718 \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

4. Applied rewrites 37.3%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. distribute-lft-out N/A

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

   8. sqrt-prod N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-/.f32 85.9

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

6. Applied rewrites 85.2%

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

7. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

9. Applied rewrites 71.5%

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

10. Taylor expanded in u1 around inf

    \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{u2} \]
11. Step-by-step derivation

12. Applied rewrites 19.5%

    \[\leadsto 6.28318530718 \cdot \color{blue}{u2} \]

13. Final simplification 19.5%

    \[\leadsto u2 \cdot 6.28318530718 \]
14. Add Preprocessing

Alternative 7: 4.3% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ -6.28318530718 \cdot u2 \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

4. Applied rewrites 36.4%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. distribute-lft-out N/A

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

   8. sqrt-prod N/A

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

   9. lift-sqrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-/.f32 85.9

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

6. Applied rewrites 85.9%

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

7. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

9. Applied rewrites 71.2%

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

10. Taylor expanded in u1 around -inf

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

12. Applied rewrites 4.3%

    \[\leadsto -6.28318530718 \cdot \color{blue}{u2} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024285 
(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))))