Toniolo and Linder, Equation (3b), real

Average Accuracy: 93.6% → 99.7%

Time: 44.5s

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 93.6%

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

2. Simplified 99.7%

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

   [Start] 93.6	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 93.6	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 93.6	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 93.6	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 99.7	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	52.5%
Cost	58580

\[\begin{array}{l} t_1 := \frac{ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Accuracy	52.5%
Cost	58580

\[\begin{array}{l} t_1 := \sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq -4 \cdot 10^{-302}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Accuracy	46.5%
Cost	52180

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\left|\sin th \cdot \frac{\sin ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 4
Accuracy	46.5%
Cost	52180

\[\begin{array}{l} t_1 := \frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\left|t_1\right|\\ \mathbf{elif}\;\sin kx \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	43.9%
Cost	45648

\[\begin{array}{l} t_1 := \frac{\sin kx}{\sin th}\\ \mathbf{if}\;\sin kx \leq -4 \cdot 10^{-64}:\\ \;\;\;\;\frac{ky}{\left|t_1\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1}\\ \end{array} \]

Alternative 6
Accuracy	46.3%
Cost	45648

\[\begin{array}{l} t_1 := \frac{ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Accuracy	37.4%
Cost	39248

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-57}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 8
Accuracy	37.4%
Cost	39248

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-57}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 9
Accuracy	37.4%
Cost	39248

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-57}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]

Alternative 10
Accuracy	41.2%
Cost	39248

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -4 \cdot 10^{-64}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]

Alternative 11
Accuracy	41.2%
Cost	39248

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -4 \cdot 10^{-64}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \end{array} \]

Alternative 12
Accuracy	66.2%
Cost	39048

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.005:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 10^{-15}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \left|\frac{\sin th}{\sin kx}\right|\\ \end{array} \]

Alternative 13
Accuracy	43.9%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -4 \cdot 10^{-64}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 14
Accuracy	44.0%
Cost	32584

\[\begin{array}{l} t_1 := \frac{\sin th}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -4 \cdot 10^{-64}:\\ \;\;\;\;\left|ky \cdot t_1\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot t_1\\ \end{array} \]

Alternative 15
Accuracy	44.0%
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -4 \cdot 10^{-64}:\\ \;\;\;\;\frac{ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \mathbf{elif}\;\sin kx \leq 10^{-203}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 16
Accuracy	75.1%
Cost	26892

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky}{t_1} \cdot th\\ \mathbf{if}\;kx \leq -0.011:\\ \;\;\;\;t_2\\ \mathbf{elif}\;kx \leq 31500:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 7 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\frac{1}{t_1}}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}\\ \end{array} \]

Alternative 17
Accuracy	73.0%
Cost	26513

\[\begin{array}{l} t_1 := \sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;kx \leq -0.042:\\ \;\;\;\;t_1\\ \mathbf{elif}\;kx \leq 31500:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 4.3 \cdot 10^{+71} \lor \neg \left(kx \leq 1.7 \cdot 10^{+247}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \end{array} \]

Alternative 18
Accuracy	73.0%
Cost	26512

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky}{t_1} \cdot th\\ \mathbf{if}\;kx \leq -0.019:\\ \;\;\;\;t_2\\ \mathbf{elif}\;kx \leq 31500:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 5 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;kx \leq 5.6 \cdot 10^{+247}:\\ \;\;\;\;\sin th \cdot \left|\frac{\sin ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \end{array} \]

Alternative 19
Accuracy	74.9%
Cost	26380

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky}{t_1} \cdot th\\ \mathbf{if}\;kx \leq -0.039:\\ \;\;\;\;t_2\\ \mathbf{elif}\;kx \leq 31500:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;kx \leq 3.4 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{t_1}\\ \end{array} \]

Alternative 20
Accuracy	31.0%
Cost	26316

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -1 \cdot 10^{-57}:\\ \;\;\;\;\left|\frac{ky \cdot th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 0.17:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \end{array} \]

Alternative 21
Accuracy	33.4%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-90}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 22
Accuracy	32.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -4.8 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.45 \cdot 10^{-93}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 23
Accuracy	29.8%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -4.8 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.6 \cdot 10^{-267}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 24
Accuracy	21.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -0.23:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-89}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 25
Accuracy	21.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -0.4:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.42 \cdot 10^{-89}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 26
Accuracy	13.2%
Cost	64

\[th \]

Reproduce

herbie shell --seed 2023140 
(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)))