Average Error: 4.2 → 0.2
Time: 33.5s
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 4.2

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

  2. Simplified0.2

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

    [Start]4.2

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

    +-commutative [=>]4.2

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

    unpow2 [=>]4.2

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

    unpow2 [=>]4.2

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

    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
Error15.4
Cost52113
\[\begin{array}{l} t_1 := \frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-9}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 0.75 \lor \neg \left(\sin ky \leq 0.998\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 2
Error15.4
Cost52113
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky \cdot th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-9}:\\ \;\;\;\;ky \cdot \frac{\sin th}{t_1}\\ \mathbf{elif}\;\sin ky \leq 0.75 \lor \neg \left(\sin ky \leq 0.998\right):\\ \;\;\;\;\frac{\sin ky}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 3
Error32.5
Cost45648
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin ky \leq 10^{-47}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 4
Error32.7
Cost39116
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq -1 \cdot 10^{-293}:\\ \;\;\;\;\frac{-ky}{\frac{-\mathsf{hypot}\left(\sin kx, \sin ky\right)}{th}}\\ \mathbf{elif}\;\sin ky \leq 10^{-47}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 5
Error32.6
Cost39116
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\frac{-ky}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(th \cdot -0.16666666666666666 + \frac{-1}{th}\right)}\\ \mathbf{elif}\;\sin ky \leq 10^{-47}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{\sin kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 6
Error35.5
Cost32584
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-89}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-112}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \end{array} \]
Alternative 7
Error21.9
Cost32516
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 2 \cdot 10^{-9}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 8
Error36.9
Cost26184
\[\begin{array}{l} \mathbf{if}\;\sin kx \leq -2 \cdot 10^{-89}:\\ \;\;\;\;\left|ky \cdot \frac{\sin th}{\sin kx}\right|\\ \mathbf{elif}\;\sin kx \leq 10^{-102}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \end{array} \]
Alternative 9
Error37.7
Cost13649
\[\begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{-105} \lor \neg \left(ky \leq 5.6 \cdot 10^{-73}\right) \land ky \leq 7.2 \cdot 10^{-48}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 10
Error37.7
Cost13648
\[\begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{elif}\;ky \leq 2.2 \cdot 10^{-72}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 8.5 \cdot 10^{-48}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 11
Error37.7
Cost13648
\[\begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;ky \leq 1.7 \cdot 10^{-72}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 9 \cdot 10^{-48}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 12
Error37.7
Cost13648
\[\begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 3 \cdot 10^{-106}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{elif}\;ky \leq 1.06 \cdot 10^{-73}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{ky \cdot \sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 13
Error40.8
Cost13516
\[\begin{array}{l} \mathbf{if}\;ky \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -1.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{elif}\;ky \leq 2.5 \cdot 10^{-103}:\\ \;\;\;\;\sin th \cdot \left|\frac{ky}{kx}\right|\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 14
Error41.9
Cost13252
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-202}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 15
Error41.8
Cost13252
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-202}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 16
Error42.7
Cost6984
\[\begin{array}{l} \mathbf{if}\;ky \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6 \cdot 10^{-202}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 17
Error43.5
Cost6728
\[\begin{array}{l} \mathbf{if}\;ky \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 8.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{th}{\frac{kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 18
Error55.5
Cost320
\[th \cdot \frac{ky}{kx} \]
Alternative 19
Error55.5
Cost320
\[\frac{th}{\frac{kx}{ky}} \]

Error

Reproduce

herbie shell --seed 2023012 
(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)))