Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.4% → 98.1%

Time: 10.5s

Alternatives: 10

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: 98.4% 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: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

4. Applied rewrites 97.9%

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

6. Applied rewrites 98.2%

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

Alternative 2: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0007999999797903001:\\ \;\;\;\;\left(\sqrt{\frac{-1}{u1 - 1}} \cdot u2\right) \cdot \left(6.28318530718 \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-u1\right) \cdot \left(\left(-u1\right) - 1\right)} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0007999999797903001)
   (* (* (sqrt (/ -1.0 (- u1 1.0))) u2) (* 6.28318530718 (sqrt u1)))
   (* (sqrt (* (- u1) (- (- u1) 1.0))) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0007999999797903001f) {
		tmp = (sqrtf((-1.0f / (u1 - 1.0f))) * u2) * (6.28318530718f * sqrtf(u1));
	} else {
		tmp = sqrtf((-u1 * (-u1 - 1.0f))) * 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 (u2 <= 0.0007999999797903001e0) then
        tmp = (sqrt(((-1.0e0) / (u1 - 1.0e0))) * u2) * (6.28318530718e0 * sqrt(u1))
    else
        tmp = sqrt((-u1 * (-u1 - 1.0e0))) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0007999999797903001))
		tmp = Float32(Float32(sqrt(Float32(Float32(-1.0) / Float32(u1 - Float32(1.0)))) * u2) * Float32(Float32(6.28318530718) * sqrt(u1)));
	else
		tmp = Float32(sqrt(Float32(Float32(-u1) * Float32(Float32(-u1) - Float32(1.0)))) * sin(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(0.0007999999797903001))
		tmp = (sqrt((single(-1.0) / (u1 - single(1.0)))) * u2) * (single(6.28318530718) * sqrt(u1));
	else
		tmp = sqrt((-u1 * (-u1 - single(1.0)))) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 7.9999998e-4

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. 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 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 97.3%

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

   9. Applied rewrites 97.6%

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

   if 7.9999998e-4 < u2

   1. Initial program 98.0%

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower--.f32 N/A

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

      3. lower-neg.f32 89.0

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

   7. Applied rewrites 89.0%

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

Alternative 3: 98.4% 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 98.2%

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

Alternative 4: 89.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0020000000949949026:\\ \;\;\;\;\left(\sqrt{\frac{-1}{u1 - 1}} \cdot u2\right) \cdot \left(6.28318530718 \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0020000000949949026)
   (* (* (sqrt (/ -1.0 (- u1 1.0))) u2) (* 6.28318530718 (sqrt u1)))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0020000000949949026f) {
		tmp = (sqrtf((-1.0f / (u1 - 1.0f))) * u2) * (6.28318530718f * sqrtf(u1));
	} 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 (u2 <= 0.0020000000949949026e0) then
        tmp = (sqrt(((-1.0e0) / (u1 - 1.0e0))) * u2) * (6.28318530718e0 * sqrt(u1))
    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 (u2 <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(sqrt(Float32(Float32(-1.0) / Float32(u1 - Float32(1.0)))) * u2) * Float32(Float32(6.28318530718) * sqrt(u1)));
	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 (u2 <= single(0.0020000000949949026))
		tmp = (sqrt((single(-1.0) / (u1 - single(1.0)))) * u2) * (single(6.28318530718) * sqrt(u1));
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00200000009

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. 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 95.7

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

   5. Applied rewrites 95.7%

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

   7. Applied rewrites 95.8%

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

   9. Applied rewrites 96.1%

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

   if 0.00200000009 < u2

   1. Initial program 98.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 76.8

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

   5. Applied rewrites 76.8%

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

Alternative 5: 81.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. 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 82.9

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

5. Applied rewrites 82.9%

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

7. Applied rewrites 82.9%

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

9. Applied rewrites 83.1%

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

Alternative 6: 81.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. 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 82.9

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

5. Applied rewrites 82.9%

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

7. Applied rewrites 83.0%

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

Alternative 7: 81.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. 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 82.9

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

5. Applied rewrites 82.9%

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

7. Applied rewrites 82.9%

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

Alternative 8: 81.7% 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 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. 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 82.9

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

5. Applied rewrites 82.9%

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

Alternative 9: 64.6% 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 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. 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 82.9

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

5. Applied rewrites 82.9%

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

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 67.0%

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

Alternative 10: 64.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. 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 82.9

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

5. Applied rewrites 82.9%

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

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 67.0%

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

10. Applied rewrites 66.9%

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

Reproduce

herbie shell --seed 2024332 
(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))))