Toniolo and Linder, Equation (3b), real

Average Error: 3.8 → 0.2

Time: 26.8s

Precision: binary64

Cost: 32384

Math

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

↓

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

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 3.8

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

2. Simplified 0.2

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

   [Start] 3.8	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 3.8	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 3.8	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 3.8	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 0.2	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Applied egg-rr 35.8

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1} \]

4. Simplified 0.2

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

   [Start] 35.8	\[ e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]
   expm1-def [=>] 0.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]
   expm1-log1p [=>] 0.3	\[ \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   *-commutative [=>] 0.3	\[ \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   associate-/r/ [<=] 0.2	\[ \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	22.1
Cost	45580

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.925:\\ \;\;\;\;\frac{\frac{\sin th}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.001:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 2
Error	22.1
Cost	45580

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.925:\\ \;\;\;\;\frac{\frac{\sin th}{ky \cdot 0.16666666666666666 + \frac{1}{ky}}}{\sin kx}\\ \mathbf{elif}\;\sin th \leq -0.001:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-11}:\\ \;\;\;\;th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 3
Error	15.0
Cost	39048

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-7}:\\ \;\;\;\;th \cdot \frac{\sin ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;ky \cdot \frac{\sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	15.0
Cost	39048

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\sin ky}{t_1}}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-5}:\\ \;\;\;\;ky \cdot \frac{\sin th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	33.7
Cost	32584

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \]

Alternative 6
Error	0.2
Cost	32384

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

Alternative 7
Error	33.7
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	33.7
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-111}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 9
Error	33.7
Cost	26184

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 10
Error	38.5
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-161}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 11
Error	43.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 9.5 \cdot 10^{-161}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 12
Error	43.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.1:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 13
Error	45.0
Cost	6728

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.15 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 14
Error	51.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.15 \cdot 10^{-107}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 15
Error	55.3
Cost	64

\[th \]

Reproduce

herbie shell --seed 2023047 
(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)))