Toniolo and Linder, Equation (3b), real

Average Error: 3.8 → 0.2

Time: 35.4s

Precision: binary64

Cost: 32384

Math

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

↓

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

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 ky) (hypot (sin ky) (sin kx))) (sin th)))

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(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

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(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

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(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

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(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
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(ky) / hypot(sin(ky), sin(kx))) * sin(th);
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[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.8

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

2. Simplified 0.2

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

   [Start] 3.8	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 3.8	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 3.8	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 3.8	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 0.2	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	36.9
Cost	38984

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	33.6
Cost	38984

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-290}:\\ \;\;\;\;\frac{\sin th}{\frac{1}{t_1}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\sin th \cdot \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	33.3
Cost	38984

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	37.4
Cost	32584

\[\begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\frac{t_1}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\left|ky \cdot t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	37.4
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\sin kx}{ky}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	36.9
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\sin kx}{ky}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	36.9
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\sin kx}{ky}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	22.0
Cost	32516

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 0.03:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Error	16.5
Cost	26764

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky}{t_1 \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{if}\;ky \leq -0.0005:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ky \leq 0.028:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{ky}}\\ \mathbf{elif}\;ky \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	16.6
Cost	26380

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \sin ky \cdot \frac{th}{t_1}\\ \mathbf{if}\;ky \leq -0.012:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ky \leq 0.045:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;ky \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	16.7
Cost	26380

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -0.0012:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;ky \leq 0.028:\\ \;\;\;\;\sin th \cdot \frac{ky}{t_1}\\ \mathbf{elif}\;ky \leq 1.5 \cdot 10^{+93}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \end{array} \]

Alternative 12
Error	16.7
Cost	26380

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;ky \leq -0.0022:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;ky \leq 0.07:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{ky}}\\ \mathbf{elif}\;ky \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \end{array} \]

Alternative 13
Error	37.4
Cost	20036

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	37.7
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{\sin th}{\sin kx}}{\frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	37.8
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	37.8
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Error	42.4
Cost	13516

\[\begin{array}{l} \mathbf{if}\;ky \leq -7 \cdot 10^{-24}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -3.95 \cdot 10^{-203}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;ky \leq 3 \cdot 10^{-119}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18
Error	42.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -7 \cdot 10^{-24}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 7.5 \cdot 10^{-120}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19
Error	43.8
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -15.6:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.3 \cdot 10^{-120}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 20
Error	49.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -15:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.75 \cdot 10^{-119}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 21
Error	55.2
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022364 
(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)))