Toniolo and Linder, Equation (3b), real

Average Error: 93.9% → 99.7%

Time: 1.0min

Precision: binary64

Cost: 32384.00

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 93.9

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

2. Simplified 99.7

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

   [Start] 93.9	\[ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   +-commutative [=>] 93.9	\[ \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]
   unpow2 [=>] 93.9	\[ \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]
   unpow2 [=>] 93.9	\[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]
   hypot-def [=>] 99.7	\[ \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

3. Applied egg-rr 43.9

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

4. Simplified 99.7

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

   [Start] 43.9	\[ e^{\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} - 1 \]
   expm1-def [=>] 99.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)\right)} \]
   expm1-log1p [=>] 99.6	\[ \color{blue}{\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \]
   *-commutative [=>] 99.6	\[ \color{blue}{\frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky} \]
   associate-/r/ [<=] 99.7	\[ \color{blue}{\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}} \]

5. Final simplification 99.7

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

Alternatives

Alternative 1
Error	54.2%
Cost	65245.00

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ t_2 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky}}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-268}:\\ \;\;\;\;t_2 \cdot t_1\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-307}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-128} \lor \neg \left(\sin ky \leq 4 \cdot 10^{-89}\right) \land \sin ky \leq 10^{-11}:\\ \;\;\;\;\sin th \cdot \left|t_1\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 2
Error	53.1%
Cost	65244.00

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ t_2 := \sin th \cdot \left|t_1\right|\\ t_3 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky}}\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-268}:\\ \;\;\;\;t_3 \cdot t_1\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-307}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 3
Error	52.6%
Cost	65244.00

\[\begin{array}{l} t_1 := \frac{ky}{\sin kx}\\ t_2 := \sin th \cdot \left|t_1\right|\\ t_3 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-269}:\\ \;\;\;\;t_3 \cdot t_1\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-296}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 4
Error	52.6%
Cost	65244.00

\[\begin{array}{l} t_1 := \sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ t_2 := \left|\sin th\right|\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-269}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{t_2}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-296}:\\ \;\;\;\;\left|\sin ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 5
Error	52.8%
Cost	58712.00

\[\begin{array}{l} t_1 := \sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-116}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{ky}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\left|\sin ky\right|}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sin th}{1 + \frac{0.5 \cdot \left(kx \cdot kx\right)}{{\sin ky}^{2}}}\\ \mathbf{elif}\;\sin ky \leq 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 6
Error	52.3%
Cost	52181.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-307}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-128} \lor \neg \left(\sin ky \leq 4 \cdot 10^{-89}\right) \land \sin ky \leq 10^{-11}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 7
Error	51.8%
Cost	52181.00

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ t_2 := \frac{ky}{\sin kx}\\ \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-296}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-128} \lor \neg \left(\sin ky \leq 4 \cdot 10^{-89}\right) \land \sin ky \leq 10^{-11}:\\ \;\;\;\;\sin th \cdot \left|t_2\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 8
Error	38.0%
Cost	39248.00

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{th \cdot ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -4 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-236}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]

Alternative 9
Error	38.0%
Cost	39248.00

\[\begin{array}{l} t_1 := \left|\sin th\right|\\ \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-55}:\\ \;\;\;\;\left|\frac{th \cdot ky}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq -4 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-236}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin kx \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]

Alternative 10
Error	74.3%
Cost	39049.00

\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin th \leq -0.1 \lor \neg \left(\sin th \leq 10^{-7}\right):\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{th}{\frac{t_1}{\sin ky}}\\ \end{array} \]

Alternative 11
Error	66.0%
Cost	39048.00

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\left|\sin ky\right|}}\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 12
Error	66.0%
Cost	39048.00

\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.05:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{\left|\sin ky\right|}}\\ \mathbf{elif}\;\sin th \leq 10^{-7}:\\ \;\;\;\;\frac{th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\frac{\sin kx}{\sin th}\right|}\\ \end{array} \]

Alternative 13
Error	99.7%
Cost	32384.00

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

Alternative 14
Error	99.6%
Cost	32384.00

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

Alternative 15
Error	42.4%
Cost	19784.00

\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 16
Error	31.5%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -9.5 \cdot 10^{+25}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.8 \cdot 10^{-227}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 17
Error	34.3%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.5 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.7 \cdot 10^{-175}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 18
Error	34.3%
Cost	6984.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.5 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.55 \cdot 10^{-175}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 19
Error	21.8%
Cost	6728.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -7.5 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \mathbf{elif}\;ky \leq 1.56 \cdot 10^{-167}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(th\right)\\ \end{array} \]

Alternative 20
Error	30.9%
Cost	6728.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -1.5 \cdot 10^{-6}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4 \cdot 10^{-226}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]

Alternative 21
Error	21.5%
Cost	584.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{-161}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 22
Error	21.5%
Cost	584.00

\[\begin{array}{l} \mathbf{if}\;ky \leq -1 \cdot 10^{-52}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.15 \cdot 10^{-162}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]

Alternative 23
Error	13.4%
Cost	64.00

\[th \]

Reproduce

herbie shell --seed 2023093 
(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)))