?

Average Accuracy: 93.6% → 99.7%
Time: 37.7s
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 94.2%

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

  2. Simplified99.6%

    \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]
    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 [=>]99.6

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

  3. Final simplification99.6%

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

Alternatives

Alternative 1
Accuracy46.3%
Cost52244
\[\begin{array}{l} t_1 := \frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin ky}\\ \end{array} \]
Alternative 2
Accuracy44.6%
Cost45780
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 3
Accuracy44.5%
Cost45780
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 4
Accuracy44.5%
Cost45780
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 5
Accuracy44.5%
Cost45780
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-97}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 6
Accuracy76.8%
Cost39176
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.0005:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin th}{t_1 \cdot \frac{1}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 7
Accuracy60.5%
Cost39116
\[\begin{array}{l} \mathbf{if}\;\sin th \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin th \leq 0.82:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \end{array} \]
Alternative 8
Accuracy60.6%
Cost39116
\[\begin{array}{l} \mathbf{if}\;\sin th \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;\sin th \leq 0.0005:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin th \leq 0.82:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sin kx}\\ \end{array} \]
Alternative 9
Accuracy99.6%
Cost32384
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
Alternative 10
Accuracy72.5%
Cost26513
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{ky \cdot \sin th}{t_1}\\ \mathbf{if}\;th \leq -0.0032:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \mathbf{elif}\;th \leq 8.5 \cdot 10^{+118} \lor \neg \left(th \leq 1.8 \cdot 10^{+222}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 11
Accuracy34.2%
Cost19652
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 12
Accuracy39.8%
Cost13780
\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;ky \leq 3.8 \cdot 10^{-129}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sin th}{\frac{\sin kx}{ky}}\\ \mathbf{elif}\;ky \leq 7.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 13
Accuracy39.6%
Cost13384
\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 14
Accuracy39.6%
Cost13384
\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 15
Accuracy34.0%
Cost13252
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 16
Accuracy33.9%
Cost13252
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 10^{-209}:\\ \;\;\;\;\frac{\sin th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 17
Accuracy32.6%
Cost6984
\[\begin{array}{l} \mathbf{if}\;ky \leq -3.2:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{-212}:\\ \;\;\;\;th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 18
Accuracy24.0%
Cost6464
\[\sin th \]
Alternative 19
Accuracy13.9%
Cost64
\[th \]

Error

Reproduce?

herbie shell --seed 2023157 
(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)))