Toniolo and Linder, Equation (3b), real

Average Error: 4.1 → 0.2

Time: 30.2s

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 4.1

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

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\right)\right)} \cdot \sin th \]

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	15.3
Cost	39176

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-15}:\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 10^{-27}:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	24.8
Cost	39048

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.002:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.002:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 3
Error	17.8
Cost	39048

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 10^{-27}:\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	36.7
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.206:\\ \;\;\;\;\frac{th \cdot ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]

Alternative 5
Error	37.3
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left({th}^{3}, -0.16666666666666666, th\right)\right)}^{2}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \]

Alternative 6
Error	37.7
Cost	26052

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-27}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	37.7
Cost	26052

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-27}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	37.4
Cost	26052

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

Alternative 9
Error	38.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.6490076276888446 \cdot 10^{-16}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.989003850130547 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	38.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.6490076276888446 \cdot 10^{-16}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.989003850130547 \cdot 10^{-28}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	38.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.6490076276888446 \cdot 10^{-16}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.989003850130547 \cdot 10^{-28}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	43.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -2.6643666526119056 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.5378391269175858 \cdot 10^{-178}:\\ \;\;\;\;\frac{th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	43.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -2.6643666526119056 \cdot 10^{-15}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.5378391269175858 \cdot 10^{-178}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	46.0
Cost	6920

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.6490076276888446 \cdot 10^{-16}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.5378391269175858 \cdot 10^{-178}:\\ \;\;\;\;{th}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	48.8
Cost	6464

\[\sin th \]

Alternative 16
Error	55.3
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022302 
(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)))