?

Average Accuracy: 94.2% → 99.7%
Time: 31.0s
Precision: binary64
Cost: 32384

?

\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
↓
\[\frac{\sin th}{\frac{\mathsf{hypot}\left(\sin ky, \sin kx\right)}{\sin ky}} \]
(FPCore (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))))
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));
}

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));
}

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))

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

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

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]

\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}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 94.2%

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

  2. Simplified99.7%

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

    [Start]94.2

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

    +-commutative [=>]94.2

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

    unpow2 [=>]94.2

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

    unpow2 [=>]94.2

    \[ \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

    hypot-def [=>]99.7

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

  3. Applied egg-rr99.7%

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

    [Start]99.7

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

    *-commutative [=>]99.7

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

    clear-num [=>]99.6

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

    un-div-inv [=>]99.7

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

  4. Final simplification99.7%

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

Alternatives

Alternative 1
Accuracy72.4%
Cost58644
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.578:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 0.0001:\\ \;\;\;\;\frac{\sin th \cdot ky}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.69:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 0.9375:\\ \;\;\;\;\frac{th \cdot \sin ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 2
Accuracy76.1%
Cost58644
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.578:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 0.0001:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{elif}\;\sin ky \leq 0.69:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 0.9375:\\ \;\;\;\;\frac{th \cdot \sin ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 3
Accuracy76.3%
Cost58644
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.578:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\sin ky \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 0.0001:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \left(ky \cdot 0.16666666666666666 + \frac{1}{ky}\right)}\\ \mathbf{elif}\;\sin ky \leq 0.69:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 0.9375:\\ \;\;\;\;\frac{th \cdot \sin ky}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 4
Accuracy75.8%
Cost39048
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.16:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-13}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
Alternative 5
Accuracy75.8%
Cost39048
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.16:\\ \;\;\;\;\frac{th \cdot \sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin kx \leq 10^{-13}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
Alternative 6
Accuracy35.2%
Cost32913
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;\sin ky \leq 8 \cdot 10^{-220} \lor \neg \left(\sin ky \leq 5 \cdot 10^{-114}\right) \land \sin ky \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sin th}{kx} \cdot \left(-ky\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 7
Accuracy35.2%
Cost32912
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;\sin ky \leq 8 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sin th}{kx} \cdot \left(-ky\right)\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sin th \cdot \left(-ky\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 8
Accuracy41.1%
Cost32912
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-130}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 8 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sin th \cdot \left(-ky\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 9
Accuracy45.9%
Cost32584
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-130}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-182}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \]
Alternative 10
Accuracy45.6%
Cost32584
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-182}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot \sin ky}{\sin ky}\\ \end{array} \]
Alternative 11
Accuracy68.2%
Cost32516
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-13}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \end{array} \]
Alternative 12
Accuracy99.6%
Cost32384
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
Alternative 13
Accuracy99.7%
Cost32384
\[\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
Alternative 14
Accuracy45.7%
Cost26184
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-130}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-178}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 15
Accuracy45.7%
Cost26184
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-130}:\\ \;\;\;\;\left|\sin th\right|\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-178}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 16
Accuracy35.1%
Cost19784
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{elif}\;\sin ky \leq 8 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 17
Accuracy30.2%
Cost13124
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-222}:\\ \;\;\;\;\sqrt{th \cdot th}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 18
Accuracy29.1%
Cost6728
\[\begin{array}{l} \mathbf{if}\;ky \leq -39000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 4.6 \cdot 10^{-222}:\\ \;\;\;\;-1 + \left(th + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 19
Accuracy19.3%
Cost841
\[\begin{array}{l} \mathbf{if}\;ky \leq -0.00365 \lor \neg \left(ky \leq 4.2 \cdot 10^{-220}\right):\\ \;\;\;\;\frac{1}{\frac{1}{th} + th \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(th + 1\right)\\ \end{array} \]
Alternative 20
Accuracy19.0%
Cost584
\[\begin{array}{l} \mathbf{if}\;ky \leq -39000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 4.8 \cdot 10^{-222}:\\ \;\;\;\;-1 + \left(th + 1\right)\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
Alternative 21
Accuracy13.2%
Cost64
\[th \]

Error

Reproduce?

herbie shell --seed 2023151 
(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)))