Toniolo and Linder, Equation (3b), real

Average Error: 3.7 → 0.2

Time: 31.2s

Precision: binary64

Cost: 38916

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \ne 0:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{t_1}\\ \end{array} \]

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
 (let* ((t_1 (hypot (sin ky) (sin kx))))
   (if (!= (sin ky) 0.0)
     (/ (sin th) (/ t_1 (sin ky)))
     (/ (* (sin ky) (sin th)) t_1))))

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) {
	double t_1 = hypot(sin(ky), sin(kx));
	double tmp;
	if (sin(ky) != 0.0) {
		tmp = sin(th) / (t_1 / sin(ky));
	} else {
		tmp = (sin(ky) * sin(th)) / t_1;
	}
	return tmp;
}

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) {
	double t_1 = Math.hypot(Math.sin(ky), Math.sin(kx));
	double tmp;
	if (Math.sin(ky) != 0.0) {
		tmp = Math.sin(th) / (t_1 / Math.sin(ky));
	} else {
		tmp = (Math.sin(ky) * Math.sin(th)) / t_1;
	}
	return tmp;
}

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):
	t_1 = math.hypot(math.sin(ky), math.sin(kx))
	tmp = 0
	if math.sin(ky) != 0.0:
		tmp = math.sin(th) / (t_1 / math.sin(ky))
	else:
		tmp = (math.sin(ky) * math.sin(th)) / t_1
	return tmp

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)
	t_1 = hypot(sin(ky), sin(kx))
	tmp = 0.0
	if (sin(ky) != 0.0)
		tmp = Float64(sin(th) / Float64(t_1 / sin(ky)));
	else
		tmp = Float64(Float64(sin(ky) * sin(th)) / t_1);
	end
	return tmp
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_2 = code(kx, ky, th)
	t_1 = hypot(sin(ky), sin(kx));
	tmp = 0.0;
	if (sin(ky) ~= 0.0)
		tmp = sin(th) / (t_1 / sin(ky));
	else
		tmp = (sin(ky) * sin(th)) / t_1;
	end
	tmp_2 = tmp;
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_] := Block[{t$95$1 = N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Unequal[N[Sin[ky], $MachinePrecision], 0.0], N[(N[Sin[th], $MachinePrecision] / N[(t$95$1 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Initial program 3.7

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

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sin ky \ne 0:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ } \end{array}} \]

3. Simplified 0.2

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sin ky \ne 0:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ } \end{array}} \]
   Proof

Alternatives

Alternative 1
Error	14.0
Cost	52244

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{if}\;\sin ky \leq -0.88:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq -0.3:\\ \;\;\;\;\frac{th}{t_1} \cdot \sin ky\\ \mathbf{elif}\;\sin ky \leq -0.02:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 0.0005:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;ky \ne 0:\\ \;\;\;\;\frac{\sin th}{\frac{t_1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	16.1
Cost	39048

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \frac{\sin th \cdot ky}{t_1}\\ \mathbf{if}\;\sin th \leq -0.005:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	21.2
Cost	38984

\[\begin{array}{l} t_1 := \frac{\sin ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{if}\;\sin kx \leq -1.9 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 2.45 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.2
Cost	32384

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

Alternative 5
Error	18.4
Cost	26580

\[\begin{array}{l} t_1 := \frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky\\ t_2 := \frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{if}\;th \leq -2.25 \cdot 10^{+112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq -1.4 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq -0.082:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 30000000:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{elif}\;th \leq 7.2 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.4
Cost	26580

\[\begin{array}{l} t_1 := \frac{\sin th}{\left|\sin kx\right|} \cdot \sin ky\\ t_2 := \frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{if}\;th \leq -1.95 \cdot 10^{+112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq -4.6 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq -0.072:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 30000000:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{elif}\;th \leq 8.5 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	33.7
Cost	26184

\[\begin{array}{l} t_1 := \frac{\sin ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{if}\;kx \leq -5 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;kx \leq 1.8 \cdot 10^{-165}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	21.0
Cost	26184

\[\begin{array}{l} t_1 := \left|\sin kx\right|\\ \mathbf{if}\;kx \leq -4.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sin th}{t_1} \cdot \sin ky\\ \mathbf{elif}\;kx \leq 4.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot \sin th\\ \end{array} \]

Alternative 9
Error	52.0
Cost	19584

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

Alternative 10
Error	47.0
Cost	19584

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

Reproduce

herbie shell --seed 2023010 
(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)))