Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 9.4s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 2: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{u1 \cdot u1 + u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9975000023841858)
     (* (sqrt (+ (* u1 u1) u1)) t_0)
     (*
      (+
       (* (fma (* 64.93939402268539 u2) u2 -19.739208802181317) (* u2 u2))
       1.0)
      (sqrt (/ u1 (- 1.0 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9975000023841858f) {
		tmp = sqrtf(((u1 * u1) + u1)) * t_0;
	} else {
		tmp = ((fmaf((64.93939402268539f * u2), u2, -19.739208802181317f) * (u2 * u2)) + 1.0f) * sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9975000023841858))
		tmp = Float32(sqrt(Float32(Float32(u1 * u1) + u1)) * t_0);
	else
		tmp = Float32(Float32(Float32(fma(Float32(Float32(64.93939402268539) * u2), u2, Float32(-19.739208802181317)) * Float32(u2 * u2)) + Float32(1.0)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.997500002

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 14.4

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

   5. Applied rewrites 12.3%

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

   7. Applied rewrites 87.4%

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

   if 0.997500002 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   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

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

   5. Applied rewrites 91.4%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{u1 \cdot u1 + u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9975000023841858)
     (* (sqrt (* (- u1 -1.0) u1)) t_0)
     (*
      (+
       (* (fma (* 64.93939402268539 u2) u2 -19.739208802181317) (* u2 u2))
       1.0)
      (sqrt (/ u1 (- 1.0 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9975000023841858f) {
		tmp = sqrtf(((u1 - -1.0f) * u1)) * t_0;
	} else {
		tmp = ((fmaf((64.93939402268539f * u2), u2, -19.739208802181317f) * (u2 * u2)) + 1.0f) * sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9975000023841858))
		tmp = Float32(sqrt(Float32(Float32(u1 - Float32(-1.0)) * u1)) * t_0);
	else
		tmp = Float32(Float32(Float32(fma(Float32(Float32(64.93939402268539) * u2), u2, Float32(-19.739208802181317)) * Float32(u2 * u2)) + Float32(1.0)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.997500002

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 13.3

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

   5. Applied rewrites 12.0%

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

   7. Applied rewrites 87.4%

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

   if 0.997500002 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   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

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9975000023841858:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \leq 0.9879999756813049:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right)\right) \cdot t\_1 + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))) (t_1 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= t_0 0.9879999756813049)
     (* (sqrt u1) t_0)
     (+
      (*
       (* (fma (* 64.93939402268539 u2) u2 -19.739208802181317) (* u2 u2))
       t_1)
      t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (t_0 <= 0.9879999756813049f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = ((fmaf((64.93939402268539f * u2), u2, -19.739208802181317f) * (u2 * u2)) * t_1) + t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9879999756813049))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = Float32(Float32(Float32(fma(Float32(Float32(64.93939402268539) * u2), u2, Float32(-19.739208802181317)) * Float32(u2 * u2)) * t_1) + t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.987999976

   1. Initial program 97.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 74.4

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

   5. Applied rewrites 74.4%

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

   if 0.987999976 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   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

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 98.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9879999756813049:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right)\right) \cdot t\_0 + t\_0 \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (+
    (* (* (fma (* 64.93939402268539 u2) u2 -19.739208802181317) (* u2 u2)) t_0)
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return ((fmaf((64.93939402268539f * u2), u2, -19.739208802181317f) * (u2 * u2)) * t_0) + t_0;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(Float32(Float32(fma(Float32(Float32(64.93939402268539) * u2), u2, Float32(-19.739208802181317)) * Float32(u2 * u2)) * t_0) + t_0)
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 79.6%

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

7. Applied rewrites 88.0%

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

8. Final simplification 88.0%

   \[\leadsto \left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \sqrt{\frac{u1}{1 - u1}} \]
9. Add Preprocessing

Alternative 6: 88.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return ((fmaf((64.93939402268539f * u2), u2, -19.739208802181317f) * (u2 * u2)) + 1.0f) * sqrtf((u1 / (1.0f - u1)));
}

Julia

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 79.6%

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

7. Applied rewrites 87.6%

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

8. Final simplification 88.0%

   \[\leadsto \left(\mathsf{fma}\left(64.93939402268539 \cdot u2, u2, -19.739208802181317\right) \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
9. Add Preprocessing

Alternative 7: 80.7% 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 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 80.0%

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

Alternative 8: 63.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 80.0%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
7. Step-by-step derivation

8. Applied rewrites 64.1%

   \[\leadsto \sqrt{u1} \]
9. Add Preprocessing

Alternative 9: 19.4% 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 99.1%

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{{\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{-0.25}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

5. Taylor expanded in u1 around inf

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

   1. lower-cos.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 20.1

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

7. Applied rewrites 20.1%

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

8. Taylor expanded in u2 around 0

   \[\leadsto 1 \]
9. Step-by-step derivation

10. Applied rewrites 19.1%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024273 
(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))))