Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.3s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot 6.28318530718\right) \cdot {\left({\left(\frac{u1 - 1}{u1}\right)}^{-2}\right)}^{0.25} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* u2 6.28318530718)) (pow (pow (/ (- u1 1.0) u1) -2.0) 0.25)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) * powf(powf(((u1 - 1.0f) / u1), -2.0f), 0.25f);
}

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 = sin((u2 * 6.28318530718e0)) * ((((u1 - 1.0e0) / u1) ** (-2.0e0)) ** 0.25e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(6.28318530718))) * ((Float32(Float32(u1 - Float32(1.0)) / u1) ^ Float32(-2.0)) ^ Float32(0.25)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) * ((((u1 - single(1.0)) / u1) ^ single(-2.0)) ^ single(0.25));
end

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{{\left({\left(\frac{u1 - 1}{u1}\right)}^{-2}\right)}^{0.25}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Final simplification 98.5%

   \[\leadsto \sin \left(u2 \cdot 6.28318530718\right) \cdot {\left({\left(\frac{u1 - 1}{u1}\right)}^{-2}\right)}^{0.25} \]
6. Add Preprocessing

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - u1}{u1}\\ {\left(t\_0 \cdot t\_0\right)}^{-0.25} \cdot \sin \left(u2 \cdot 6.28318530718\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ (- 1.0 u1) u1)))
   (* (pow (* t_0 t_0) -0.25) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f - u1) / u1;
	return powf((t_0 * t_0), -0.25f) * 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
    real(4) :: t_0
    t_0 = (1.0e0 - u1) / u1
    code = ((t_0 * t_0) ** (-0.25e0)) * sin((u2 * 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) - u1) / u1)
	return Float32((Float32(t_0 * t_0) ^ Float32(-0.25)) * sin(Float32(u2 * Float32(6.28318530718))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = (single(1.0) - u1) / u1;
	tmp = ((t_0 * t_0) ^ single(-0.25)) * sin((u2 * single(6.28318530718)));
end

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

   \[\leadsto {\left(\frac{1 - u1}{u1} \cdot \frac{1 - u1}{u1}\right)}^{-0.25} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
6. Add Preprocessing

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\left(1 - u1\right) - 0 \cdot \frac{u1 - 1}{u1}}{\frac{1 - u1}{u1} \cdot \left(1 - u1\right)}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (/
    (- (- 1.0 u1) (* 0.0 (/ (- u1 1.0) u1)))
    (* (/ (- 1.0 u1) u1) (- 1.0 u1))))
  (sin (* u2 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((((1.0f - u1) - (0.0f * ((u1 - 1.0f) / u1))) / (((1.0f - u1) / u1) * (1.0f - u1)))) * 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((((1.0e0 - u1) - (0.0e0 * ((u1 - 1.0e0) / u1))) / (((1.0e0 - u1) / u1) * (1.0e0 - u1)))) * sin((u2 * 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(Float32(1.0) - u1) - Float32(Float32(0.0) * Float32(Float32(u1 - Float32(1.0)) / u1))) / Float32(Float32(Float32(Float32(1.0) - u1) / u1) * Float32(Float32(1.0) - u1)))) * sin(Float32(u2 * Float32(6.28318530718))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((single(1.0) - u1) - (single(0.0) * ((u1 - single(1.0)) / u1))) / (((single(1.0) - u1) / u1) * (single(1.0) - u1)))) * sin((u2 * single(6.28318530718)));
end

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.4%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Final simplification 98.4%

   \[\leadsto \sqrt{\frac{\left(1 - u1\right) - 0 \cdot \frac{u1 - 1}{u1}}{\frac{1 - u1}{u1} \cdot \left(1 - u1\right)}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
6. Add Preprocessing

Alternative 4: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 6: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.014299999922513962:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{\frac{-1}{u1 - 1} \cdot \left(\frac{1 - u1}{u1} \cdot \left(u1 - 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.014299999922513962)
   (*
    (* u2 6.28318530718)
    (sqrt (/ -1.0 (* (/ -1.0 (- u1 1.0)) (* (/ (- 1.0 u1) u1) (- u1 1.0))))))
   (* (sqrt u1) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.014299999922513962f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((-1.0f / ((-1.0f / (u1 - 1.0f)) * (((1.0f - u1) / u1) * (u1 - 1.0f)))));
	} else {
		tmp = sqrtf(u1) * sinf((u2 * 6.28318530718f));
	}
	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 * 6.28318530718e0) <= 0.014299999922513962e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt(((-1.0e0) / (((-1.0e0) / (u1 - 1.0e0)) * (((1.0e0 - u1) / u1) * (u1 - 1.0e0)))))
    else
        tmp = sqrt(u1) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.014299999922513962))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(-1.0) / Float32(Float32(Float32(-1.0) / Float32(u1 - Float32(1.0))) * Float32(Float32(Float32(Float32(1.0) - u1) / u1) * Float32(u1 - Float32(1.0)))))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.014299999922513962))
		tmp = (u2 * single(6.28318530718)) * sqrt((single(-1.0) / ((single(-1.0) / (u1 - single(1.0))) * (((single(1.0) - u1) / u1) * (u1 - single(1.0))))));
	else
		tmp = sqrt(u1) * sin((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0142999999

   1. Initial program 98.6%

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

   4. Applied rewrites 98.6%

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. Applied rewrites 98.4%

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

      1. lift-/.f32 N/A

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

      2. div-inv N/A

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

      3. lift-/.f32 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 N/A

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in u2 around 0

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

       1. lower-*.f32 96.5

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

   11. Applied rewrites 96.5%

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

   if 0.0142999999 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.014299999922513962:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{\frac{-1}{u1 - 1} \cdot \left(\frac{1 - u1}{u1} \cdot \left(u1 - 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.4%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \frac{u1 - 1}{u1} - \left(1 - u1\right)}{\left(1 - u1\right) \cdot \frac{u1 - 1}{u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-times N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 N/A

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in u2 around 0

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

    1. lower-*.f32 85.8

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

11. Applied rewrites 85.8%

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

12. Final simplification 85.8%

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

Alternative 8: 81.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. Applied rewrites 85.7%

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

7. Applied rewrites 85.8%

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

8. Final simplification 85.8%

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

Alternative 9: 81.3% accurate, 3.9× speedup

Math

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

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(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(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 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. Applied rewrites 85.7%

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

6. Final simplification 85.7%

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

Alternative 10: 64.5% accurate, 6.4× speedup

Math

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

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(sqrt(u1) * Float32(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 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. Applied rewrites 85.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 68.9%

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

9. Final simplification 68.9%

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

Alternative 11: 64.5% 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 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. Applied rewrites 85.7%

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

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 68.8%

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

10. Applied rewrites 68.8%

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

Reproduce

herbie shell --seed 2024308 
(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))))