Trowbridge-Reitz Sample, near normal, slope_x

Average Error: 0.4 → 0.4

Time: 20.1s

Precision: binary64

Cost: 13376

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

Math

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

↓

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary64
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

↓

(FPCore (cosTheta_i u1 u2)
 :precision binary64
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

double code(double cosTheta_i, double u1, double u2) {
	return sqrt((u1 / (1.0 - u1))) * cos((6.28318530718 * u2));
}

↓

double code(double cosTheta_i, double u1, double u2) {
	return sqrt((u1 / (1.0 - u1))) * cos((6.28318530718 * u2));
}

Fortran

real(8) function code(costheta_i, u1, u2)
    real(8), intent (in) :: costheta_i
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt((u1 / (1.0d0 - u1))) * cos((6.28318530718d0 * u2))
end function

↓

real(8) function code(costheta_i, u1, u2)
    real(8), intent (in) :: costheta_i
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt((u1 / (1.0d0 - u1))) * cos((6.28318530718d0 * u2))
end function

Java

public static double code(double cosTheta_i, double u1, double u2) {
	return Math.sqrt((u1 / (1.0 - u1))) * Math.cos((6.28318530718 * u2));
}

↓

public static double code(double cosTheta_i, double u1, double u2) {
	return Math.sqrt((u1 / (1.0 - u1))) * Math.cos((6.28318530718 * u2));
}

Python

def code(cosTheta_i, u1, u2):
	return math.sqrt((u1 / (1.0 - u1))) * math.cos((6.28318530718 * u2))

↓

def code(cosTheta_i, u1, u2):
	return math.sqrt((u1 / (1.0 - u1))) * math.cos((6.28318530718 * u2))

Julia

function code(cosTheta_i, u1, u2)
	return Float64(sqrt(Float64(u1 / Float64(1.0 - u1))) * cos(Float64(6.28318530718 * u2)))
end

↓

function code(cosTheta_i, u1, u2)
	return Float64(sqrt(Float64(u1 / Float64(1.0 - u1))) * cos(Float64(6.28318530718 * u2)))
end

MATLAB

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

↓

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

Wolfram

code[cosTheta$95$i_, u1_, u2_] := N[(N[Sqrt[N[(u1 / N[(1.0 - u1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(6.28318530718 * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[cosTheta$95$i_, u1_, u2_] := N[(N[Sqrt[N[(u1 / N[(1.0 - u1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(6.28318530718 * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.4

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

Alternatives

Alternative 1
Error	8.2
Cost	13636

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0042:\\ \;\;\;\;\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + u1\right) \cdot u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 2
Error	11.8
Cost	13380

\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.048:\\ \;\;\;\;\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7232

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

Alternative 4
Error	26.0
Cost	6980

\[\sqrt{\begin{array}{l} \mathbf{if}\;u1 \ne 0:\\ \;\;\;\;\frac{1}{\frac{1 - u1}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{u1}{1 - u1}\\ \end{array}} \]

Alternative 5
Error	26.0
Cost	6848

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

Alternative 6
Error	33.7
Cost	6720

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

Alternative 7
Error	26.0
Cost	6720

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

Alternative 8
Error	40.1
Cost	6464

\[\sqrt{u1} \]

Reproduce

herbie shell --seed 2023010 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary64
  :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))))