Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 0.3 → 0.3

Time: 11.5s

Precision: binary32

Cost: 10336

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

Math

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

↓

\[\begin{array}{l} t_0 := 1 - u1 \cdot u1\\ \sqrt{\frac{\frac{u1}{t_0}}{\frac{1 - u1}{t_0}}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \end{array} \]

FPCore

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

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u1 u1))))
   (*
    (sqrt (/ (/ u1 t_0) (/ (- 1.0 u1) t_0)))
    (cos (sqrt (* (* u2 u2) 39.47841760436263))))))

C

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

↓

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f - (u1 * u1);
	return sqrtf(((u1 / t_0) / ((1.0f - u1) / t_0))) * cosf(sqrtf(((u2 * u2) * 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 = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

↓

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 = sqrt(((u1 / t_0) / ((1.0e0 - u1) / t_0))) * cos(sqrt(((u2 * u2) * 39.47841760436263e0)))
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

↓

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) - Float32(u1 * u1))
	return Float32(sqrt(Float32(Float32(u1 / t_0) / Float32(Float32(Float32(1.0) - u1) / t_0))) * cos(sqrt(Float32(Float32(u2 * u2) * Float32(39.47841760436263)))))
end

MATLAB

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

↓

function tmp = code(cosTheta_i, u1, u2)
	t_0 = single(1.0) - (u1 * u1);
	tmp = sqrt(((u1 / t_0) / ((single(1.0) - u1) / t_0))) * cos(sqrt(((u2 * u2) * single(39.47841760436263))));
end

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.3

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

3. Simplified 0.3

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)} \]
   Proof

   [Start] 0.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
   *-commutative [=>] 0.3	\[ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right) \]

4. Applied egg-rr 0.4

   \[\leadsto \sqrt{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{-1}}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

5. Applied egg-rr 0.3

   \[\leadsto \sqrt{\color{blue}{\frac{\frac{u1}{1 - u1 \cdot u1} \cdot \left(1 - u1 \cdot u1\right)}{1 - u1}}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

6. Simplified 0.3

   \[\leadsto \sqrt{\color{blue}{\frac{\frac{u1}{1 - u1 \cdot u1}}{\frac{1 - u1}{1 - u1 \cdot u1}}}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]
   Proof

   [Start] 0.3	\[ \sqrt{\frac{\frac{u1}{1 - u1 \cdot u1} \cdot \left(1 - u1 \cdot u1\right)}{1 - u1}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]
   associate-/l* [=>] 0.3	\[ \sqrt{\color{blue}{\frac{\frac{u1}{1 - u1 \cdot u1}}{\frac{1 - u1}{1 - u1 \cdot u1}}}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

7. Final simplification 0.3

   \[\leadsto \sqrt{\frac{\frac{u1}{1 - u1 \cdot u1}}{\frac{1 - u1}{1 - u1 \cdot u1}}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	9952

\[\cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]

Alternative 2
Error	1.2
Cost	6820

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3
Error	2.0
Cost	6692

\[\begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.3199999928474426:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	6688

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

Alternative 5
Error	4.0
Cost	3616

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

Alternative 6
Error	9.3
Cost	3360

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

Alternative 7
Error	6.7
Cost	3360

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

Alternative 8
Error	11.9
Cost	3232

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023066 
(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))))