Toniolo and Linder, Equation (3b), real

Average Error: 4.1 → 0.2

Time: 42.8s

Precision: binary64

Cost: 32384

Math

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

↓

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

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 kx) (sin ky)) (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(kx), sin(ky)) / 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(kx), Math.sin(ky)) / 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(kx), math.sin(ky)) / 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(kx), sin(ky)) / 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(kx), sin(ky)) / 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[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $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

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

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	45448

\[\begin{array}{l} t_1 := \frac{\sin th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;\sin th \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	25.0
Cost	39116

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.01:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.01:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.56:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 3
Error	25.0
Cost	39116

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.01:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{elif}\;\sin th \leq 0.01:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.56:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \end{array} \]

Alternative 4
Error	33.9
Cost	32712

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.25:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin ky}{\sin th \cdot \sin ky}}\\ \end{array} \]

Alternative 5
Error	33.9
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.25:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	33.9
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.25:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	33.9
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.25:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	16.5
Cost	26376

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{1}{\frac{t_1}{\sin th \cdot ky}}\\ \mathbf{if}\;th \leq -61295.825541198275:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 2.1652950089479233 \cdot 10^{-8}:\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	33.8
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	33.8
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.1:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	38.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -763584070.4473275:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 4.9088262223171556 \cdot 10^{-142}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	38.2
Cost	13384

\[\begin{array}{l} \mathbf{if}\;ky \leq -763584070.4473275:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq 4.9088262223171556 \cdot 10^{-142}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	41.3
Cost	12996

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.9706148345888325 \cdot 10^{-48}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;ky \leq -6.114590931688939 \cdot 10^{-177}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;ky \leq 4.9088262223171556 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	41.5
Cost	7116

\[\begin{array}{l} \mathbf{if}\;ky \leq -176719460024702.16:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -2.863015790571112 \cdot 10^{-134}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 4.9088262223171556 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 15
Error	43.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -545124499867.15314:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.321836815120619 \cdot 10^{-152}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	43.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -545124499867.15314:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.321836815120619 \cdot 10^{-152}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Error	44.2
Cost	6860

\[\begin{array}{l} \mathbf{if}\;ky \leq -176719460024702.16:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -3.1496720699400483 \cdot 10^{-174}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 2.321836815120619 \cdot 10^{-152}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18
Error	50.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.1496720699400483 \cdot 10^{-174}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 2.321836815120619 \cdot 10^{-152}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \end{array} \]

Alternative 19
Error	50.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.1496720699400483 \cdot 10^{-174}:\\ \;\;\;\;-th\\ \mathbf{elif}\;ky \leq 2.321836815120619 \cdot 10^{-152}:\\ \;\;\;\;\left(th + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 20
Error	53.2
Cost	260

\[\begin{array}{l} \mathbf{if}\;ky \leq -4.0824603007665985 \cdot 10^{-296}:\\ \;\;\;\;-th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 21
Error	55.3
Cost	64

\[th \]

Reproduce

herbie shell --seed 2022313 
(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)))