Toniolo and Linder, Equation (3b), real

Average Error: 4.0 → 0.2

Time: 30.9s

Precision: binary64

Cost: 32384

Math

\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

↓

\[\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

↓

(FPCore (kx ky th)
 :precision binary64
 (/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky))))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

↓

double code(double kx, double ky, double th) {
	return sin(th) / (hypot(sin(kx), sin(ky)) / sin(ky));
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

↓

public static double code(double kx, double ky, double th) {
	return Math.sin(th) / (Math.hypot(Math.sin(kx), Math.sin(ky)) / Math.sin(ky));
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

↓

def code(kx, ky, th):
	return math.sin(th) / (math.hypot(math.sin(kx), math.sin(ky)) / math.sin(ky))

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

↓

function code(kx, ky, th)
	return Float64(sin(th) / Float64(hypot(sin(kx), sin(ky)) / sin(ky)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

↓

function tmp = code(kx, ky, th)
	tmp = sin(th) / (hypot(sin(kx), sin(ky)) / sin(ky));
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

↓

code[kx_, ky_, th_] := N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 4.0

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

3. Final simplification 0.2

   \[\leadsto \frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

Alternatives

Alternative 1
Error	24.9
Cost	39432

\[\begin{array}{l} t_1 := \frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin th \leq 0.93:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	25.0
Cost	39116

\[\begin{array}{l} t_1 := \frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{if}\;\sin th \leq -0.02:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 0.01:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.93:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.2
Cost	32384

\[\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]

Alternative 4
Error	34.2
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky} + ky \cdot \frac{0.5}{kx}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	34.7
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	34.3
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	34.3
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	34.3
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-48}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Error	42.2
Cost	13656

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;ky \leq -219.7734271665244:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -4.095199025788004 \cdot 10^{-157}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky} + 0.5 \cdot \frac{ky}{kx}}\\ \mathbf{elif}\;ky \leq -5.7878754234540105 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.501651687734702 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sin th}{ky \cdot \frac{0.5}{kx} + \frac{kx}{ky}}\\ \mathbf{elif}\;ky \leq 1.866124329417162 \cdot 10^{+77}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.461917350554937 \cdot 10^{+207}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	42.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;ky \leq -219.7734271665244:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.501651687734702 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sin th}{ky \cdot \frac{0.5}{kx} + \frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	45.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -219.7734271665244:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.790320758224962 \cdot 10^{-113}:\\ \;\;\;\;\left(\sin th + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	46.3
Cost	6920

\[\begin{array}{l} \mathbf{if}\;ky \leq -219.7734271665244:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 9.536693539020358 \cdot 10^{-244}:\\ \;\;\;\;-0.16666666666666666 \cdot {th}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	49.1
Cost	6464

\[\sin th \]

Alternative 14
Error	55.0
Cost	576

\[\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666} \]

Alternative 15
Error	55.5
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022308 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))