Toniolo and Linder, Equation (3b), real

Average Error: 3.9 → 0.2

Time: 32.6s

Precision: binary64

Cost: 32384

Math

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

↓

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

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 ky) (sin kx)) (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(ky), sin(kx)) / 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(ky), Math.sin(kx)) / 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(ky), math.sin(kx)) / 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(ky), sin(kx)) / 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(ky), sin(kx)) / 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[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.9

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

2. Simplified 0.2

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

   [Start] 3.9	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 3.9	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 3.9	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 3.9	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 0.2	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	35.3
Cost	58713

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-153}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-301}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-176} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-66}\right) \land \sin ky \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	35.9
Cost	52181

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\left|\sin th \cdot ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-301}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-176} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-66}\right) \land \sin ky \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	35.4
Cost	52181

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-174}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-301}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-176} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-66}\right) \land \sin ky \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	35.4
Cost	52181

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-174}:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-301}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-176} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-66}\right) \land \sin ky \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	40.2
Cost	45781

\[\begin{array}{l} t_1 := \sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-179}:\\ \;\;\;\;th \cdot \frac{\left|ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-214} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-66}\right) \land \sin ky \leq 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	40.2
Cost	45781

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-93}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-179}:\\ \;\;\;\;th \cdot \frac{\left|ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-214} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-66}\right) \land \sin ky \leq 10^{-37}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	40.2
Cost	45780

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-93}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-179}:\\ \;\;\;\;th \cdot \frac{\left|ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-214}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-37}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	40.2
Cost	45780

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-179}:\\ \;\;\;\;th \cdot \frac{\left|ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-214}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-37}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Error	40.2
Cost	45780

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-93}:\\ \;\;\;\;\frac{1}{\sin kx} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-179}:\\ \;\;\;\;th \cdot \frac{\left|ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-214}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-37}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	39.2
Cost	45780

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-93}:\\ \;\;\;\;\frac{1}{\sin kx} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-174}:\\ \;\;\;\;th \cdot \frac{\left|ky\right|}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-37}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	14.9
Cost	39048

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.04:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 0.01:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	22.0
Cost	32516

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

Alternative 13
Error	0.2
Cost	32384

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

Alternative 14
Error	36.5
Cost	20445

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{1}{\sin kx} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{elif}\;ky \leq -3.6 \cdot 10^{-175}:\\ \;\;\;\;th \cdot \frac{\left|ky\right|}{\sin kx}\\ \mathbf{elif}\;ky \leq -2.5 \cdot 10^{-306}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-173} \lor \neg \left(ky \leq 4.2 \cdot 10^{-66}\right) \land ky \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	39.4
Cost	13649

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.65 \cdot 10^{-212} \lor \neg \left(ky \leq 5 \cdot 10^{-66}\right) \land ky \leq 1.85 \cdot 10^{-34}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	42.0
Cost	13516

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -2.6 \cdot 10^{-306}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{elif}\;ky \leq 2.25 \cdot 10^{-209}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Error	44.0
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-214}:\\ \;\;\;\;\sin ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18
Error	43.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.1:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.5 \cdot 10^{-209}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19
Error	44.1
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.4:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{-209}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 20
Error	55.7
Cost	320

\[th \cdot \frac{ky}{kx} \]

Alternative 21
Error	55.7
Cost	320

\[\frac{ky}{\frac{kx}{th}} \]

Reproduce

herbie shell --seed 2023053 
(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)))