Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 7.9s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

Alternative 2: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot t\_0 \leq 0.024000000208616257:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= (* (sqrt (/ u1 (- 1.0 u1))) t_0) 0.024000000208616257)
     (* (sqrt (* u1 (+ 1.0 u1))) t_0)
     (*
      (sqrt (* u1 (/ 1.0 (- 1.0 u1))))
      (+
       (* (* (- (* (* u2 u2) 64.93939402268539) 19.739208802181317) u2) u2)
       1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if ((sqrtf((u1 / (1.0f - u1))) * t_0) <= 0.024000000208616257f) {
		tmp = sqrtf((u1 * (1.0f + u1))) * t_0;
	} else {
		tmp = sqrtf((u1 * (1.0f / (1.0f - u1)))) * ((((((u2 * u2) * 64.93939402268539f) - 19.739208802181317f) * u2) * u2) + 1.0f);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if ((sqrt((u1 / (1.0e0 - u1))) * t_0) <= 0.024000000208616257e0) then
        tmp = sqrt((u1 * (1.0e0 + u1))) * t_0
    else
        tmp = sqrt((u1 * (1.0e0 / (1.0e0 - u1)))) * ((((((u2 * u2) * 64.93939402268539e0) - 19.739208802181317e0) * u2) * u2) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * t_0) <= Float32(0.024000000208616257))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + u1))) * t_0);
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(Float32(Float32(Float32(Float32(Float32(u2 * u2) * Float32(64.93939402268539)) - Float32(19.739208802181317)) * u2) * u2) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if ((sqrt((u1 / (single(1.0) - u1))) * t_0) <= single(0.024000000208616257))
		tmp = sqrt((u1 * (single(1.0) + u1))) * t_0;
	else
		tmp = sqrt((u1 * (single(1.0) / (single(1.0) - u1)))) * ((((((u2 * u2) * single(64.93939402268539)) - single(19.739208802181317)) * u2) * u2) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 0.0240000002

   1. Initial program 98.8%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-+.f32 97.9

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

   7. Applied rewrites 97.9%

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

   if 0.0240000002 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.3%

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2}} + 1\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}}, {u2}^{2}, 1\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, \color{blue}{u2 \cdot u2}, 1\right) \]

      9. lower-*.f32 85.5

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, \color{blue}{u2 \cdot u2}, 1\right) \]

   7. Applied rewrites 84.4%

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, u2 \cdot u2, 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.8%

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.11999999731779099)
   (*
    (sqrt (* u1 (/ 1.0 (- 1.0 u1))))
    (+
     (* (* (- (* (* u2 u2) 64.93939402268539) 19.739208802181317) u2) u2)
     1.0))
   (* (sqrt u1) (cos (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.11999999731779099f) {
		tmp = sqrtf((u1 * (1.0f / (1.0f - u1)))) * ((((((u2 * u2) * 64.93939402268539f) - 19.739208802181317f) * u2) * u2) + 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf((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 (u2 <= 0.11999999731779099e0) then
        tmp = sqrt((u1 * (1.0e0 / (1.0e0 - u1)))) * ((((((u2 * u2) * 64.93939402268539e0) - 19.739208802181317e0) * u2) * u2) + 1.0e0)
    else
        tmp = sqrt(u1) * cos((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.11999999731779099))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(Float32(Float32(Float32(Float32(Float32(u2 * u2) * Float32(64.93939402268539)) - Float32(19.739208802181317)) * u2) * u2) + Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.11999999731779099))
		tmp = sqrt((u1 * (single(1.0) / (single(1.0) - u1)))) * ((((((u2 * u2) * single(64.93939402268539)) - single(19.739208802181317)) * u2) * u2) + single(1.0));
	else
		tmp = sqrt(u1) * cos((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.119999997

   1. Initial program 99.3%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2}} + 1\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}}, {u2}^{2}, 1\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, \color{blue}{u2 \cdot u2}, 1\right) \]

      9. lower-*.f32 87.1

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, \color{blue}{u2 \cdot u2}, 1\right) \]

   7. Applied rewrites 86.7%

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, u2 \cdot u2, 1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + \color{blue}{1}\right) \]

   if 0.119999997 < u2

   1. Initial program 95.7%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.4

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

   5. Applied rewrites 76.4%

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

Alternative 4: 85.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0010999999940395355:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \mathsf{fma}\left(u1 - -1, u1, 1\right)} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0010999999940395355)
   (sqrt (/ u1 (- 1.0 u1)))
   (*
    (sqrt (* u1 (fma (- u1 -1.0) u1 1.0)))
    (+
     (* (* (- (* (* u2 u2) 64.93939402268539) 19.739208802181317) u2) u2)
     1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0010999999940395355f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf((u1 * fmaf((u1 - -1.0f), u1, 1.0f))) * ((((((u2 * u2) * 64.93939402268539f) - 19.739208802181317f) * u2) * u2) + 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0010999999940395355))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(sqrt(Float32(u1 * fma(Float32(u1 - Float32(-1.0)), u1, Float32(1.0)))) * Float32(Float32(Float32(Float32(Float32(Float32(u2 * u2) * Float32(64.93939402268539)) - Float32(19.739208802181317)) * u2) * u2) + Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0011

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower--.f32 96.7

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

   5. Applied rewrites 96.7%

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

   if 0.0011 < u2

   1. Initial program 97.9%

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2}} + 1\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}}, {u2}^{2}, 1\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, \color{blue}{u2 \cdot u2}, 1\right) \]

      9. lower-*.f32 41.3

         \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, \color{blue}{u2 \cdot u2}, 1\right) \]

   7. Applied rewrites 41.0%

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, u2 \cdot u2, 1\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(1 + u1 \cdot \left(1 + u1\right)\right)}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(1 + u1\right) + 1\right)}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \left(\color{blue}{\left(1 + u1\right) \cdot u1} + 1\right)} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(1 + u1, u1, 1\right)}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(\color{blue}{u1 + 1}, u1, 1\right)} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      5. rgt-mult-inverse N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1 + \color{blue}{u1 \cdot \frac{1}{u1}}, u1, 1\right)} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      6. fp-cancel-sign-sub N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(\color{blue}{u1 - \left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{1}{u1}}, u1, 1\right)} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1 - \color{blue}{\left(\mathsf{neg}\left(u1 \cdot \frac{1}{u1}\right)\right)}, u1, 1\right)} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      8. rgt-mult-inverse N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1 - \left(\mathsf{neg}\left(\color{blue}{1}\right)\right), u1, 1\right)} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1 - \color{blue}{-1}, u1, 1\right)} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \]

      10. lower--.f32 39.7

          \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(\color{blue}{u1 - -1}, u1, 1\right)} \cdot \mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, u2 \cdot u2, 1\right) \]

   10. Applied rewrites 39.7%

       \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1 - -1, u1, 1\right)}} \cdot \mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, u2 \cdot u2, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 57.2%

       \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1 - -1, u1, 1\right)} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* u1 (/ 1.0 (- 1.0 u1))))
  (+
   (* (* (- (* (* u2 u2) 64.93939402268539) 19.739208802181317) u2) u2)
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f / (1.0f - u1)))) * ((((((u2 * u2) * 64.93939402268539f) - 19.739208802181317f) * u2) * u2) + 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 = sqrt((u1 * (1.0e0 / (1.0e0 - u1)))) * ((((((u2 * u2) * 64.93939402268539e0) - 19.739208802181317e0) * u2) * u2) + 1.0e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(Float32(Float32(Float32(Float32(Float32(u2 * u2) * Float32(64.93939402268539)) - Float32(19.739208802181317)) * u2) * u2) + Float32(1.0)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) / (single(1.0) - u1)))) * ((((((u2 * u2) * single(64.93939402268539)) - single(19.739208802181317)) * u2) * u2) + single(1.0));
end

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) \cdot {u2}^{2}} + 1\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}}, {u2}^{2}, 1\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

   6. unpow2 N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \color{blue}{\left(u2 \cdot u2\right)} - \frac{98696044010906577398881}{5000000000000000000000}, {u2}^{2}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(u2 \cdot u2\right) - \frac{98696044010906577398881}{5000000000000000000000}, \color{blue}{u2 \cdot u2}, 1\right) \]

   9. lower-*.f32 80.3

      \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, \color{blue}{u2 \cdot u2}, 1\right) \]

7. Applied rewrites 80.3%

   \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \color{blue}{\mathsf{fma}\left(64.93939402268539 \cdot \left(u2 \cdot u2\right) - 19.739208802181317, u2 \cdot u2, 1\right)} \]
8. Step-by-step derivation

9. Applied rewrites 91.4%

   \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \left(\left(\left(\left(u2 \cdot u2\right) \cdot 64.93939402268539 - 19.739208802181317\right) \cdot u2\right) \cdot u2 + \color{blue}{1}\right) \]
10. Add Preprocessing

Alternative 6: 80.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.5

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

5. Applied rewrites 80.5%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
6. Add Preprocessing

Alternative 7: 63.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, u1, 1\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (fma 0.5 u1 1.0) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf(0.5f, u1, 1.0f) * sqrtf(u1);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(Float32(0.5), u1, Float32(1.0)) * sqrt(u1))
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.5

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

5. Applied rewrites 80.5%

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

7. Applied rewrites 80.0%

   \[\leadsto {\left(1 - u1\right)}^{-0.5} \cdot \color{blue}{\sqrt{u1}} \]

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 63.7%

    \[\leadsto \mathsf{fma}\left(0.5, u1, 1\right) \cdot \sqrt{\color{blue}{u1}} \]
11. Add Preprocessing

Alternative 8: 62.0% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.5

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

5. Applied rewrites 80.5%

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

7. Applied rewrites 62.1%

   \[\leadsto \sqrt{\frac{u1}{u1 + 1}} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + -1 \cdot u1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 62.0%

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

Alternative 9: 14.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{u1} - 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (- (/ 0.5 u1) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return (0.5f / 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 = (0.5e0 / u1) - 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(0.5) / u1) - Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(0.5) / u1) - single(1.0);
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.5

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

5. Applied rewrites 80.5%

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

7. Applied rewrites 62.1%

   \[\leadsto \sqrt{\frac{u1}{u1 + 1}} \]

8. Taylor expanded in u1 around -inf

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

10. Applied rewrites 14.2%

    \[\leadsto \frac{0.5}{u1} - \color{blue}{1} \]
11. Add Preprocessing

Alternative 10: 4.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.5}{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (- 1.0 (/ 0.5 u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f - (0.5f / 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 = 1.0e0 - (0.5e0 / u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) - Float32(Float32(0.5) / u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) - (single(0.5) / u1);
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.5

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

5. Applied rewrites 80.5%

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

7. Applied rewrites 62.1%

   \[\leadsto \sqrt{\frac{u1}{u1 + 1}} \]

8. Taylor expanded in u1 around inf

   \[\leadsto 1 - \color{blue}{\frac{1}{2} \cdot \frac{1}{u1}} \]
9. Step-by-step derivation

10. Applied rewrites 4.2%

    \[\leadsto 1 - \color{blue}{\frac{0.5}{u1}} \]
11. Add Preprocessing

Alternative 11: 4.0% accurate, 135.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 -1.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return -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 = -1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower--.f32 80.5

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

5. Applied rewrites 80.5%

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

7. Applied rewrites 62.1%

   \[\leadsto \sqrt{\frac{u1}{u1 + 1}} \]

8. Taylor expanded in u1 around -inf

   \[\leadsto {\left(\sqrt{-1}\right)}^{\color{blue}{2}} \]
9. Step-by-step derivation

10. Applied rewrites 4.0%

    \[\leadsto -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))