Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 7.3s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 98.2

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

4. Applied rewrites 98.2%

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

5. Final simplification 98.2%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 98.2

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

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]
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 98.2

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

6. Applied rewrites 98.2%

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

7. Final simplification 98.2%

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

Alternative 3: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0026000000070780516:\\ \;\;\;\;\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0026000000070780516)
   (* (* (sqrt (/ u1 (- 1.0 u1))) 6.28318530718) u2)
   (* (sqrt (* (+ u1 1.0) u1)) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0026000000070780516f) {
		tmp = (sqrtf((u1 / (1.0f - u1))) * 6.28318530718f) * u2;
	} else {
		tmp = sqrtf(((u1 + 1.0f) * u1)) * sinf((6.28318530718f * u2));
	}
	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) <= 0.0026000000070780516e0) then
        tmp = (sqrt((u1 / (1.0e0 - u1))) * 6.28318530718e0) * u2
    else
        tmp = sqrt(((u1 + 1.0e0) * u1)) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0026000000070780516))
		tmp = Float32(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(6.28318530718)) * u2);
	else
		tmp = Float32(sqrt(Float32(Float32(u1 + Float32(1.0)) * u1)) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0026000000070780516))
		tmp = (sqrt((u1 / (single(1.0) - u1))) * single(6.28318530718)) * u2;
	else
		tmp = sqrt(((u1 + single(1.0)) * u1)) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00260000001

   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 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.00260000001 < (*.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

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0026000000070780516:\\ \;\;\;\;\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 5: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009499999694526196:\\ \;\;\;\;\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009499999694526196)
   (* (* (sqrt (/ u1 (- 1.0 u1))) 6.28318530718) u2)
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009499999694526196f) {
		tmp = (sqrtf((u1 / (1.0f - u1))) * 6.28318530718f) * u2;
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	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) <= 0.009499999694526196e0) then
        tmp = (sqrt((u1 / (1.0e0 - u1))) * 6.28318530718e0) * u2
    else
        tmp = sqrt(u1) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009499999694526196))
		tmp = Float32(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(6.28318530718)) * u2);
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.009499999694526196))
		tmp = (sqrt((u1 / (single(1.0) - u1))) * single(6.28318530718)) * u2;
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00949999969

   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 96.5

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

   5. Applied rewrites 96.5%

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

   if 0.00949999969 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 6: 81.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
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 80.6

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

5. Applied rewrites 80.6%

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

Alternative 7: 81.6% accurate, 3.9× speedup

Math

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

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(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * 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 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-+.f32 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 80.5

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

7. Applied rewrites 80.5%

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

9. Applied rewrites 80.5%

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

Alternative 8: 81.7% accurate, 3.9× 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 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-+.f32 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 80.5

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

7. Applied rewrites 80.5%

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

9. Applied rewrites 80.4%

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

Alternative 9: 64.9% 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 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 80.6

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

5. Applied rewrites 80.6%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 63.6%

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

Alternative 10: 64.9% accurate, 6.4× speedup

Math

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

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(Float32(sqrt(u1) * 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 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 80.6

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

5. Applied rewrites 80.6%

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

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 63.5%

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

Alternative 11: 64.8% accurate, 6.4× speedup

Math

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

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(sqrt(u1) * Float32(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 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 80.6

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

5. Applied rewrites 80.6%

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

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 63.5%

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

10. Applied rewrites 63.5%

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

Reproduce

herbie shell --seed 2024298 
(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))))