Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 11.6s

Alternatives: 17

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.6499999761581421:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right), \left(\mathsf{fma}\left(-76.70585975309672 \cdot u2, u2, 81.6052492761019\right) \cdot t\_0\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* u2 6.28318530718) 0.6499999761581421)
     (*
      (fma
       t_0
       (fma (* -41.341702240407926 u2) u2 6.28318530718)
       (*
        (* (fma (* -76.70585975309672 u2) u2 81.6052492761019) t_0)
        (* (* u2 u2) (* u2 u2))))
      u2)
     (* (sqrt (fma (fma u1 u1 u1) u1 u1)) (sin (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.6499999761581421f) {
		tmp = fmaf(t_0, fmaf((-41.341702240407926f * u2), u2, 6.28318530718f), ((fmaf((-76.70585975309672f * u2), u2, 81.6052492761019f) * t_0) * ((u2 * u2) * (u2 * u2)))) * u2;
	} else {
		tmp = sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1)) * sinf((u2 * 6.28318530718f));
	}
	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(u2 * Float32(6.28318530718)) <= Float32(0.6499999761581421))
		tmp = Float32(fma(t_0, fma(Float32(Float32(-41.341702240407926) * u2), u2, Float32(6.28318530718)), Float32(Float32(fma(Float32(Float32(-76.70585975309672) * u2), u2, Float32(81.6052492761019)) * t_0) * Float32(Float32(u2 * u2) * Float32(u2 * u2)))) * u2);
	else
		tmp = Float32(sqrt(fma(fma(u1, u1, u1), u1, u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.649999976

   1. Initial program 98.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f32 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-sqrt.f32 88.2

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

   10. Applied rewrites 98.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}}, \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right), \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-76.70585975309672 \cdot u2, u2, 81.6052492761019\right)\right)\right) \cdot u2} \]

   if 0.649999976 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.6499999761581421:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}}, \mathsf{fma}\left(-41.341702240407926 \cdot u2, u2, 6.28318530718\right), \left(\mathsf{fma}\left(-76.70585975309672 \cdot u2, u2, 81.6052492761019\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right)\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))