Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 8.3s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

   1. lift--.f32 N/A

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

   2. flip-- N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f32 N/A

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

   5. metadata-eval N/A

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

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

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

   7. distribute-neg-in N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. sqr-neg-rev N/A

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

   10. lower-+.f32 N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. distribute-neg-in N/A

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

   14. *-rgt-identity N/A

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

   15. distribute-lft-neg-in N/A

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

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

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

   17. *-rgt-identity N/A

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

   18. lower--.f32 N/A

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

   19. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 76.7%

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

6. Applied rewrites 98.5%

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

7. Final simplification 98.5%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

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

Alternative 4: 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

Derivation

1. Initial program 98.4%

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

Alternative 5: 94.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.20000000298023224f) {
		tmp = (((u2 * u2) * -41.341702240407926f) - -6.28318530718f) * (sqrtf((u1 / (1.0f - u1))) * 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) <= 0.20000000298023224e0) then
        tmp = (((u2 * u2) * (-41.341702240407926e0)) - (-6.28318530718e0)) * (sqrt((u1 / (1.0e0 - u1))) * 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(0.20000000298023224))
		tmp = Float32(Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) - Float32(-6.28318530718)) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * 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(0.20000000298023224))
		tmp = (((u2 * u2) * single(-41.341702240407926)) - single(-6.28318530718)) * (sqrt((u1 / (single(1.0) - u1))) * 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) < 0.200000003

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 92.3

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   8. Applied rewrites 91.9%

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

   10. Applied rewrites 97.7%

       \[\leadsto \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) - -6.28318530718\right)\right) \cdot u2 \]
   11. Step-by-step derivation

   12. Applied rewrites 97.8%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 75.9

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

   5. Applied rewrites 75.9%

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

Alternative 6: 88.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 83.1%

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

10. Applied rewrites 89.9%

    \[\leadsto \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) - -6.28318530718\right)\right) \cdot u2 \]
11. Step-by-step derivation

12. Applied rewrites 90.0%

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

Alternative 7: 88.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return ((((u2 * u2) * -41.341702240407926f) - -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 = ((((u2 * u2) * (-41.341702240407926e0)) - (-6.28318530718e0)) * u2) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 82.8%

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

10. Applied rewrites 89.9%

    \[\leadsto \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) - -6.28318530718\right)\right) \cdot u2 \]
11. Step-by-step derivation

12. Applied rewrites 89.9%

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

Alternative 8: 88.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (sqrtf((u1 / (1.0f - u1))) * (((-41.341702240407926f * u2) * u2) - -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))) * ((((-41.341702240407926e0) * u2) * u2) - (-6.28318530718e0))) * u2
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 82.8%

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

10. Applied rewrites 89.9%

    \[\leadsto \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) - -6.28318530718\right)\right) \cdot u2 \]
11. Step-by-step derivation

12. Applied rewrites 89.9%

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

Alternative 9: 88.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (sqrtf((u1 / (1.0f - u1))) * ((-41.341702240407926f * (u2 * u2)) - -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))) * (((-41.341702240407926e0) * (u2 * u2)) - (-6.28318530718e0))) * u2
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 82.8%

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

10. Applied rewrites 89.9%

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

Alternative 10: 81.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 82.8%

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

10. Applied rewrites 83.1%

    \[\leadsto \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \color{blue}{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)} \]
11. Add Preprocessing

Alternative 11: 81.3% accurate, 3.9× speedup

Math

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

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(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(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 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

Alternative 12: 81.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 82.8%

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

9. Taylor expanded in u2 around 0

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

11. Applied rewrites 83.1%

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

Alternative 13: 64.7% 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 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 69.4%

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

Alternative 14: 64.6% 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 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 83.1

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

5. Applied rewrites 83.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 69.4%

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

10. Applied rewrites 69.4%

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

Reproduce

herbie shell --seed 2024326 
(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))))