Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.4% → 98.4%

Time: 10.3s

Alternatives: 13

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: 98.4% 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 98.4%

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

Alternative 2: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.001500000013038516)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.001500000013038516f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.001500000013038516e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.001500000013038516))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.001500000013038516))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00150000001

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 98.1%

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

      1. associate-*r* 98.4%

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

   5. Simplified 98.4%

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

      1. add-sqr-sqrt 97.9%

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

      2. sqrt-unprod 98.4%

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

      3. *-commutative 98.4%

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

      4. *-commutative 98.4%

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

      5. swap-sqr 98.1%

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

      6. add-sqr-sqrt 98.3%

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

      7. metadata-eval 98.8%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{39.47841760436263}} \cdot u2 \]

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}} \cdot u2 \]

   if 0.00150000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) 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 84.5%

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

      1. +-commutative 84.5%

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

   5. Simplified 84.5%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.012000000104308128)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.012000000104308128f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	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) <= 0.012000000104308128e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.012000000104308128))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.012000000104308128))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0120000001

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 96.5%

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

      1. associate-*r* 96.7%

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

   5. Simplified 96.7%

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

      1. add-sqr-sqrt 96.3%

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

      2. sqrt-unprod 96.7%

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

      3. *-commutative 96.7%

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

      4. *-commutative 96.7%

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

      5. swap-sqr 96.5%

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

      6. add-sqr-sqrt 96.7%

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

      7. metadata-eval 97.1%

         \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{39.47841760436263}} \cdot u2 \]

   7. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}} \cdot u2 \]

   if 0.0120000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

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

   3. Taylor expanded in u1 around 0 72.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
}

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 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
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 81.3%

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

   1. associate-*r* 81.4%

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

5. Simplified 81.4%

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

   1. add-sqr-sqrt 81.1%

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

   2. sqrt-unprod 81.4%

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

   3. *-commutative 81.4%

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

   4. *-commutative 81.4%

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

   5. swap-sqr 81.2%

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

   6. add-sqr-sqrt 81.4%

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

   7. metadata-eval 81.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{39.47841760436263}} \cdot u2 \]

7. Applied egg-rr 81.7%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}} \cdot u2 \]

8. Final simplification 81.7%

   \[\leadsto u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263} \]
9. Add Preprocessing

Alternative 5: 81.4% accurate, 1.9× speedup

Math

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

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(6.28318530718) * Float32(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 81.3%

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

Alternative 6: 73.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt((u1 * (u1 + single(1.0)))));
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 81.3%

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

4. Taylor expanded in u1 around 0 73.6%

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

   1. +-commutative 86.3%

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

6. Simplified 73.6%

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

7. Final simplification 73.6%

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

Alternative 7: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = u2 * (6.28318530718e0 * sqrt(u1))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(6.28318530718) * sqrt(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 81.3%

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

   1. associate-*r* 81.4%

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

5. Simplified 81.4%

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

6. Taylor expanded in u1 around 0 64.9%

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

   1. *-commutative 64.9%

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

8. Simplified 64.9%

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

9. Final simplification 64.9%

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

Alternative 8: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * sqrt(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 81.3%

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

4. Taylor expanded in u1 around 0 73.6%

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

   1. +-commutative 73.6%

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

   2. distribute-lft-in 73.6%

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

   3. *-rgt-identity 73.6%

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

   4. fma-define 73.6%

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

6. Simplified 73.6%

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

7. Taylor expanded in u1 around 0 64.9%

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

   1. *-commutative 64.9%

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

   2. associate-*l* 64.9%

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

   3. *-commutative 64.9%

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

9. Simplified 64.9%

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

10. Final simplification 64.9%

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

Alternative 9: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt(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 81.3%

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

4. Taylor expanded in u1 around 0 64.9%

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

5. Final simplification 64.9%

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

Alternative 10: 20.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u2 \cdot \left(u1 \cdot \left(1 + \frac{0.5}{u1}\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (* u1 (+ 1.0 (/ 0.5 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * (u1 * (1.0f + (0.5f / u1))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * (u1 * (1.0e0 + (0.5e0 / u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) / u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * (u1 * (single(1.0) + (single(0.5) / 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 81.3%

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

4. Taylor expanded in u1 around 0 73.6%

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

   1. +-commutative 73.6%

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

   2. distribute-lft-in 73.6%

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

   3. *-rgt-identity 73.6%

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

   4. fma-define 73.6%

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

6. Simplified 73.6%

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

7. Taylor expanded in u1 around inf 20.2%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left(u1 \cdot \left(1 + 0.5 \cdot \frac{1}{u1}\right)\right)} \cdot u2\right) \]
8. Step-by-step derivation

   1. associate-*r/ 20.2%

      \[\leadsto 6.28318530718 \cdot \left(\left(u1 \cdot \left(1 + \color{blue}{\frac{0.5 \cdot 1}{u1}}\right)\right) \cdot u2\right) \]

   2. metadata-eval 20.2%

      \[\leadsto 6.28318530718 \cdot \left(\left(u1 \cdot \left(1 + \frac{\color{blue}{0.5}}{u1}\right)\right) \cdot u2\right) \]

9. Simplified 20.2%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left(u1 \cdot \left(1 + \frac{0.5}{u1}\right)\right)} \cdot u2\right) \]

10. Final simplification 20.2%

    \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \left(u1 \cdot \left(1 + \frac{0.5}{u1}\right)\right)\right) \]
11. Add Preprocessing

Alternative 11: 20.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u1 \cdot \left(u2 + 0.5 \cdot \frac{u2}{u1}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u1 (+ u2 (* 0.5 (/ u2 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u1 * (u2 + (0.5f * (u2 / u1))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u1 * (u2 + (0.5e0 * (u2 / u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u1 * Float32(u2 + Float32(Float32(0.5) * Float32(u2 / u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u1 * (u2 + (single(0.5) * (u2 / 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 81.3%

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

4. Taylor expanded in u1 around 0 73.6%

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

   1. +-commutative 73.6%

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

   2. distribute-lft-in 73.6%

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

   3. *-rgt-identity 73.6%

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

   4. fma-define 73.6%

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

6. Simplified 73.6%

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

7. Taylor expanded in u1 around inf 20.2%

   \[\leadsto 6.28318530718 \cdot \color{blue}{\left(u1 \cdot \left(u2 + 0.5 \cdot \frac{u2}{u1}\right)\right)} \]
8. Add Preprocessing

Alternative 12: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (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 81.3%

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

4. Taylor expanded in u1 around 0 73.6%

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

   1. +-commutative 73.6%

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

   2. distribute-lft-in 73.6%

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

   3. *-rgt-identity 73.6%

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

   4. fma-define 73.6%

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

6. Simplified 73.6%

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

7. Taylor expanded in u1 around inf 19.2%

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

Alternative 13: 4.7% accurate, 41.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (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 = (u1 * u2) * (-6.28318530718e0)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (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 81.3%

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

4. Taylor expanded in u1 around 0 73.6%

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

   1. +-commutative 73.6%

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

   2. distribute-lft-in 73.6%

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

   3. *-rgt-identity 73.6%

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

   4. fma-define 73.6%

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

6. Simplified 73.6%

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

7. Taylor expanded in u1 around -inf 4.9%

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

8. Final simplification 4.9%

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

Reproduce

herbie shell --seed 2024150 
(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))))