Toniolo and Linder, Equation (3b), real

Average Error: 3.8 → 0.3

Time: 24.4s

Precision: binary64

Cost: 32384

Math

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

↓

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

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

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

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

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

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

Derivation

1. Initial program 3.8

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

2. Simplified 2.4

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

3. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\sin ky \cdot \left(\sin th \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} \]

4. Applied egg-rr 0.3

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

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	37.6
Cost	52180

\[\begin{array}{l} t_1 := \frac{\sin ky \cdot \sin th}{\sin ky}\\ t_2 := \frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;\sin kx \leq -0.605:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin kx \leq -3 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq -1 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin kx \leq -3 \cdot 10^{-281}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 2
Error	16.0
Cost	45580

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t_1}\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 0.05:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{ky}}\\ \mathbf{elif}\;\sin ky \leq 0.54:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	22.5
Cost	32516

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	0.3
Cost	32384

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

Alternative 5
Error	37.6
Cost	26052

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-76}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	37.1
Cost	26052

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-76}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	37.1
Cost	26052

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-76}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	38.5
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -71684.1162285722:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.307659785243442 \cdot 10^{-77}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Error	38.0
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -71684.1162285722:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.307659785243442 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	38.0
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -71684.1162285722:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.307659785243442 \cdot 10^{-77}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	38.0
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -71684.1162285722:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.307659785243442 \cdot 10^{-77}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	48.7
Cost	6464

\[\sin th \]

Alternative 13
Error	55.0
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022225 
(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)))