?

Average Error: 3.9 → 0.2
Time: 39.2s
Precision: binary64
Cost: 32384

?

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

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(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

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(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

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(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
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(ky) / hypot(sin(ky), sin(kx))) * sin(th);
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[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

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

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 3.9

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

  2. Simplified0.2

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

    [Start]3.9

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

    +-commutative [=>]3.9

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

    unpow2 [=>]3.9

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

    unpow2 [=>]3.9

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

    hypot-def [=>]0.2

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

  3. Final simplification0.2

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

Alternatives

Alternative 1
Error36.5
Cost45780
\[\begin{array}{l} t_1 := \frac{ky \cdot \sin th}{\sin kx}\\ t_2 := \sqrt{{\sin th}^{2}}\\ \mathbf{if}\;\sin ky \leq -6.8 \cdot 10^{-50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq -2 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 10^{-307}:\\ \;\;\;\;\sin th \cdot \frac{-ky}{kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 2
Error34.1
Cost45712
\[\begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{th}}\\ \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\sin kx}{ky}}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 3 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{elif}\;\sin kx \leq 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
Alternative 3
Error33.4
Cost45712
\[\begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{th}}\\ t_2 := \sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\left|t_2\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 3 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{elif}\;\sin kx \leq 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error33.4
Cost45712
\[\begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(kx, \sin ky\right)}{th}}\\ \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-50}:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\sin kx}{\sin ky}}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin kx \leq 3 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \mathbf{elif}\;\sin kx \leq 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
Alternative 5
Error36.2
Cost39116
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0105:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\sin kx}{ky}}\right|\\ \mathbf{elif}\;\sin kx \leq -2 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\sin th}{\frac{kx}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 10^{-101}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
Alternative 6
Error17.3
Cost39048
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -0.0105:\\ \;\;\;\;\left|\frac{\sin th}{\frac{\sin kx}{\sin ky}}\right|\\ \mathbf{elif}\;\sin kx \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
Alternative 7
Error38.9
Cost32716
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-230}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-307}:\\ \;\;\;\;\sin th \cdot \frac{-ky}{kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 8
Error39.2
Cost32716
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -1 \cdot 10^{-230}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-307}:\\ \;\;\;\;\sin th \cdot \frac{-ky}{kx}\\ \mathbf{elif}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 9
Error34.8
Cost32584
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -5 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;\left|\sin th \cdot \frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 10
Error15.7
Cost26897
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\frac{\sin ky}{\frac{1}{th} + th \cdot 0.16666666666666666}}{t_1}\\ \mathbf{if}\;ky \leq -0.000305:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ky \leq 0.0016:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{elif}\;ky \leq 1.6 \cdot 10^{+68} \lor \neg \left(ky \leq 2.3 \cdot 10^{+142}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \end{array} \]
Alternative 11
Error15.9
Cost26512
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t_1}\\ \mathbf{if}\;ky \leq -0.00025:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ky \leq 0.0155:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{elif}\;ky \leq 9 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;ky \leq 6.2 \cdot 10^{+142}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 12
Error15.8
Cost26512
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{\sin ky \cdot th}{t_1}\\ \mathbf{if}\;ky \leq -0.00072:\\ \;\;\;\;t_2\\ \mathbf{elif}\;ky \leq 0.00155:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{elif}\;ky \leq 3.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\sin ky}{\frac{1}{th}}}{t_1}\\ \mathbf{elif}\;ky \leq 1.92 \cdot 10^{+146}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 13
Error15.6
Cost26249
\[\begin{array}{l} \mathbf{if}\;th \leq -0.0045 \lor \neg \left(th \leq 0.0062\right):\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \end{array} \]
Alternative 14
Error15.8
Cost26248
\[\begin{array}{l} \mathbf{if}\;th \leq -0.014:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;th \leq 50000:\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \end{array} \]
Alternative 15
Error40.3
Cost26184
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sin th}{kx} \cdot \left(-\sin ky\right)\\ \mathbf{elif}\;\sin kx \leq 10^{-77}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
Alternative 16
Error40.3
Cost26184
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\sin th}{\frac{kx}{\sin ky}}\\ \mathbf{elif}\;\sin kx \leq 10^{-77}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
Alternative 17
Error40.3
Cost26184
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sin ky}{\frac{-kx}{\sin th}}\\ \mathbf{elif}\;\sin kx \leq 10^{-77}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
Alternative 18
Error39.1
Cost13648
\[\begin{array}{l} t_1 := \sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{if}\;ky \leq -6.9 \cdot 10^{+16}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -7.8 \cdot 10^{-233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;ky \leq 1.15 \cdot 10^{-307}:\\ \;\;\;\;\sin th \cdot \frac{-ky}{kx}\\ \mathbf{elif}\;ky \leq 2.6 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 19
Error42.1
Cost13316
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{-ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 20
Error42.1
Cost13316
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;\frac{-\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 21
Error42.4
Cost13252
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 22
Error42.4
Cost13252
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-144}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 23
Error44.3
Cost6728
\[\begin{array}{l} \mathbf{if}\;ky \leq -6 \cdot 10^{+16}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.24 \cdot 10^{-141}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 24
Error49.9
Cost584
\[\begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{-50}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.45 \cdot 10^{-134}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
Alternative 25
Error49.9
Cost584
\[\begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{-50}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 9.4 \cdot 10^{-141}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
Alternative 26
Error49.9
Cost584
\[\begin{array}{l} \mathbf{if}\;ky \leq -6.5 \cdot 10^{-50}:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.65 \cdot 10^{-139}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
Alternative 27
Error55.0
Cost64
\[th \]

Error

Reproduce?

herbie shell --seed 2023025 
(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)))