Toniolo and Linder, Equation (3b), real

Average Error: 3.8 → 0.2

Time: 30.9s

Precision: binary64

Cost: 32384

Math

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

↓

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

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	39.2
Cost	58712

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\sin th \cdot t_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-246}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{elif}\;\sin ky \leq 10^{-40}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.1205:\\ \;\;\;\;th \cdot \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	39.0
Cost	58712

\[\begin{array}{l} t_1 := \frac{\sin ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\sin th \cdot t_1\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-246}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-40}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 0.1205:\\ \;\;\;\;th \cdot \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	39.0
Cost	52312

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ t_2 := \sin th \cdot t_1\\ t_3 := th \cdot \left|t_1\right|\\ t_4 := \frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-67}:\\ \;\;\;\;\left|t_4\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-286}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-246}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-48}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	38.4
Cost	45780

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ t_2 := th \cdot \left|t_1\right|\\ \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\sin th \cdot t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	14.7
Cost	39560

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	42.5
Cost	39249

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-286}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-127} \lor \neg \left(\sin ky \leq 2 \cdot 10^{-48}\right) \land \sin ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	14.8
Cost	39048

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	36.9
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-205}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \end{array} \]

Alternative 9
Error	22.4
Cost	32516

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	39.4
Cost	13912

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ t_2 := th \cdot \left|t_1\right|\\ t_3 := \sin th \cdot t_1\\ \mathbf{if}\;ky \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.22 \cdot 10^{-284}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;ky \leq 4 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{-131}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;ky \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.4 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	39.3
Cost	13912

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ t_2 := th \cdot \left|t_1\right|\\ t_3 := \sin th \cdot t_1\\ \mathbf{if}\;ky \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 7.8 \cdot 10^{-286}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;ky \leq 3.2 \cdot 10^{-243}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ky \leq 2.16 \cdot 10^{-131}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;ky \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{elif}\;ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	42.4
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-183}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	43.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.7 \cdot 10^{-137}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	44.2
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.8 \cdot 10^{-184}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	50.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2.4 \cdot 10^{-176}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 16
Error	55.3
Cost	64

\[th \]

Reproduce

herbie shell --seed 2023027 
(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)))