Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.2%

Time: 4.4s

Alternatives: 16

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

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 1.0× speedup

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

Math

\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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 (fma -6.28318530718 u2 1.5707963705062866))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(fmaf(-6.28318530718f, u2, 1.5707963705062866f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(fma(Float32(-6.28318530718), u2, Float32(1.5707963705062866))))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right) \]

4. Evaluated real constant 99.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 1.0× speedup

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

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \leq 0.017999999225139618:\\ \;\;\;\;t\_0 + -19.739208802181317 \cdot \left({u2}^{2} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
  (if (<= u2 0.017999999225139618)
    (+ t_0 (* -19.739208802181317 (* (pow u2 2.0) t_0)))
    (* (sqrt (fma u1 u1 u1)) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (u2 <= 0.017999999225139618f) {
		tmp = t_0 + (-19.739208802181317f * (powf(u2, 2.0f) * t_0));
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (u2 <= Float32(0.017999999225139618))
		tmp = Float32(t_0 + Float32(Float32(-19.739208802181317) * Float32((u2 ^ Float32(2.0)) * t_0)));
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0179999992

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   if 0.0179999992 < u2

   1. Initial program 99.0%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 86.8%

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

   6. Applied rewrites 86.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.2% accurate, 0.6× speedup

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

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9934999942779541:\\ \;\;\;\;\sin \left(1.5707963705062866 + -6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -19.739208802181317 \cdot \left({u2}^{2} \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
  (if (<= (cos (* 6.28318530718 u2)) 0.9934999942779541)
    (* (sin (+ 1.5707963705062866 (* -6.28318530718 u2))) (sqrt u1))
    (+ t_0 (* -19.739208802181317 (* (pow u2 2.0) t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9934999942779541f) {
		tmp = sinf((1.5707963705062866f + (-6.28318530718f * u2))) * sqrtf(u1);
	} else {
		tmp = t_0 + (-19.739208802181317f * (powf(u2, 2.0f) * t_0));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    if (cos((6.28318530718e0 * u2)) <= 0.9934999942779541e0) then
        tmp = sin((1.5707963705062866e0 + ((-6.28318530718e0) * u2))) * sqrt(u1)
    else
        tmp = t_0 + ((-19.739208802181317e0) * ((u2 ** 2.0e0) * t_0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9934999942779541))
		tmp = Float32(sin(Float32(Float32(1.5707963705062866) + Float32(Float32(-6.28318530718) * u2))) * sqrt(u1));
	else
		tmp = Float32(t_0 + Float32(Float32(-19.739208802181317) * Float32((u2 ^ Float32(2.0)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = single(0.0);
	if (cos((single(6.28318530718) * u2)) <= single(0.9934999942779541))
		tmp = sin((single(1.5707963705062866) + (single(-6.28318530718) * u2))) * sqrt(u1);
	else
		tmp = t_0 + (single(-19.739208802181317) * ((u2 ^ single(2.0)) * t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.993499994

   1. Initial program 99.0%

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

   3. Applied rewrites 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right) \]

   4. Evaluated real constant 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sin \left(\frac{13176795}{8388608} + \frac{-314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.5%

      \[\leadsto \sin \left(1.5707963705062866 + -6.28318530718 \cdot u2\right) \cdot \sqrt{u1} \]

   if 0.993499994 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.2% accurate, 0.6× speedup

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

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9934999942779541:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -19.739208802181317 \cdot \left({u2}^{2} \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
  (if (<= (cos (* 6.28318530718 u2)) 0.9934999942779541)
    (* (sqrt u1) (sin (fma -6.28318530718 u2 1.5707963705062866)))
    (+ t_0 (* -19.739208802181317 (* (pow u2 2.0) t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9934999942779541f) {
		tmp = sqrtf(u1) * sinf(fmaf(-6.28318530718f, u2, 1.5707963705062866f));
	} else {
		tmp = t_0 + (-19.739208802181317f * (powf(u2, 2.0f) * t_0));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9934999942779541))
		tmp = Float32(sqrt(u1) * sin(fma(Float32(-6.28318530718), u2, Float32(1.5707963705062866))));
	else
		tmp = Float32(t_0 + Float32(Float32(-19.739208802181317) * Float32((u2 ^ Float32(2.0)) * t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.993499994

   1. Initial program 99.0%

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

   3. Applied rewrites 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 0.5 \cdot \pi\right)\right) \]

   4. Evaluated real constant 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 74.5%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-6.28318530718, u2, 1.5707963705062866\right)\right) \]

   if 0.993499994 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.1% accurate, 0.6× speedup

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

Math

\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \leq 0.9934999942779541:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + -19.739208802181317 \cdot \left({u2}^{2} \cdot t\_1\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (cos (* 6.28318530718 u2)))
       (t_1 (sqrt (/ u1 (- 1.0 u1)))))
  (if (<= t_0 0.9934999942779541)
    (* (sqrt u1) t_0)
    (+ t_1 (* -19.739208802181317 (* (pow u2 2.0) t_1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (t_0 <= 0.9934999942779541f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = t_1 + (-19.739208802181317f * (powf(u2, 2.0f) * t_1));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    t_1 = sqrt((u1 / (1.0e0 - u1)))
    if (t_0 <= 0.9934999942779541e0) then
        tmp = sqrt(u1) * t_0
    else
        tmp = t_1 + ((-19.739208802181317e0) * ((u2 ** 2.0e0) * t_1))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9934999942779541))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = Float32(t_1 + Float32(Float32(-19.739208802181317) * Float32((u2 ^ Float32(2.0)) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	t_1 = sqrt((u1 / (single(1.0) - u1)));
	tmp = single(0.0);
	if (t_0 <= single(0.9934999942779541))
		tmp = sqrt(u1) * t_0;
	else
		tmp = t_1 + (single(-19.739208802181317) * ((u2 ^ single(2.0)) * t_1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.993499994

   1. Initial program 99.0%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 74.4%

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

   if 0.993499994 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.1% accurate, 0.6× speedup

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

Math

\[\begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9934999942779541:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (cos (* 6.28318530718 u2))))
  (if (<= t_0 0.9934999942779541)
    (* (sqrt u1) t_0)
    (*
     (sqrt (/ u1 (- 1.0 u1)))
     (+ 1.0 (* -19.739208802181317 (pow u2 2.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9934999942779541f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * powf(u2, 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9934999942779541e0) then
        tmp = sqrt(u1) * t_0
    else
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 ** 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9934999942779541))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * (u2 ^ Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9934999942779541))
		tmp = sqrt(u1) * t_0;
	else
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 ^ single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.993499994

   1. Initial program 99.0%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 74.4%

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

   if 0.993499994 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.4% accurate, 1.4× speedup

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

Math

\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
 (+ 1.0 (* -19.739208802181317 (pow u2 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (-19.739208802181317f * powf(u2, 2.0f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((-19.739208802181317e0) * (u2 ** 2.0e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * (u2 ^ Float32(2.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (single(-19.739208802181317) * (u2 ^ single(2.0))));
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \]
3. Step-by-step derivation

4. Applied rewrites 88.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + -19.739208802181317 \cdot {u2}^{2}\right) \]
5. Add Preprocessing

Alternative 8: 88.1% accurate, 1.9× speedup

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

Math

\[\frac{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317, \sqrt{u1}, \sqrt{u1}\right)}{\sqrt{\left|u1 - 1\right|}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (/
 (fma (* (* u2 u2) -19.739208802181317) (sqrt u1) (sqrt u1))
 (sqrt (fabs (- u1 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.6%

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

4. Taylor expanded in u2 around 0

   \[\leadsto \frac{\sqrt{\left|u1\right|} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
5. Step-by-step derivation

6. Applied rewrites 88.1%

   \[\leadsto \frac{\sqrt{\left|u1\right|} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
7. Step-by-step derivation

8. Applied rewrites 88.1%

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

Alternative 9: 88.1% accurate, 1.9× speedup

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

Math

\[\frac{\mathsf{fma}\left(\left(-19.739208802181317 \cdot u2\right) \cdot u2, \sqrt{u1}, \sqrt{u1}\right)}{\sqrt{\left|u1 - 1\right|}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (/
 (fma (* (* -19.739208802181317 u2) u2) (sqrt u1) (sqrt u1))
 (sqrt (fabs (- u1 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.6%

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

4. Taylor expanded in u2 around 0

   \[\leadsto \frac{\sqrt{\left|u1\right|} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
5. Step-by-step derivation

6. Applied rewrites 88.1%

   \[\leadsto \frac{\sqrt{\left|u1\right|} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
7. Step-by-step derivation

8. Applied rewrites 88.1%

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

10. Applied rewrites 88.1%

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

Alternative 10: 88.1% accurate, 1.9× speedup

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

Math

\[\frac{\mathsf{fma}\left(\sqrt{u1} \cdot u2, -19.739208802181317 \cdot u2, \sqrt{u1}\right)}{\sqrt{\left|u1 - 1\right|}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (/
 (fma (* (sqrt u1) u2) (* -19.739208802181317 u2) (sqrt u1))
 (sqrt (fabs (- u1 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.6%

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

4. Taylor expanded in u2 around 0

   \[\leadsto \frac{\sqrt{\left|u1\right|} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
5. Step-by-step derivation

6. Applied rewrites 88.1%

   \[\leadsto \frac{\sqrt{\left|u1\right|} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
7. Step-by-step derivation

8. Applied rewrites 88.1%

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

Alternative 11: 88.1% accurate, 2.1× speedup

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

Math

\[\frac{\mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right) \cdot \sqrt{u1}}{\sqrt{\left|u1 - 1\right|}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (/
 (* (fma (* u2 u2) -19.739208802181317 1.0) (sqrt u1))
 (sqrt (fabs (- u1 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.6%

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

4. Taylor expanded in u2 around 0

   \[\leadsto \frac{\sqrt{\left|u1\right|} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
5. Step-by-step derivation

6. Applied rewrites 88.1%

   \[\leadsto \frac{\sqrt{\left|u1\right|} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
7. Step-by-step derivation

8. Applied rewrites 88.1%

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

10. Applied rewrites 88.1%

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

Alternative 12: 83.3% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.0005920000257901847:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317, \sqrt{u1}, \sqrt{u1}\right)}{\sqrt{\left|-1\right|}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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)))
  (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
  (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.0005920000257901847)
    (/
     (fma (* (* u2 u2) -19.739208802181317) (sqrt u1) (sqrt u1))
     (sqrt (fabs -1.0)))
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((t_0 * cosf((6.28318530718f * u2))) <= 0.0005920000257901847f) {
		tmp = fmaf(((u2 * u2) * -19.739208802181317f), sqrtf(u1), sqrtf(u1)) / sqrtf(fabsf(-1.0f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(6.28318530718) * u2))) <= Float32(0.0005920000257901847))
		tmp = Float32(fma(Float32(Float32(u2 * u2) * Float32(-19.739208802181317)), sqrt(u1), sqrt(u1)) / sqrt(abs(Float32(-1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 5.92000026e-4

   1. Initial program 99.0%

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in u2 around 0

      \[\leadsto \frac{\sqrt{\left|u1\right|} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
   5. Step-by-step derivation

   6. Applied rewrites 88.1%

      \[\leadsto \frac{\sqrt{\left|u1\right|} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\left|u1\right|}\right)}{\sqrt{\left|u1 - 1\right|}} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.1%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 67.8%

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

   if 5.92000026e-4 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.0%

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

   2. Taylor expanded in u2 around 0

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

   4. Applied rewrites 80.1%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 80.1% accurate, 5.3× speedup

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

Math

\[\sqrt{\frac{u1}{1 - u1}} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 80.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 14: 71.7% accurate, 5.7× speedup

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

Math

\[\sqrt{u1 \cdot \left(1 + u1\right)} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 71.7%

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

Alternative 15: 63.3% accurate, 16.5× speedup

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

Math

\[\sqrt{u1} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :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))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 80.1%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 63.3%

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

Reproduce

herbie shell --seed 2025359 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))