Average Error: 3.9 → 0.2
Time: 27.0s
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
    (*.f64 (/.f64 (sin.f64 ky) (hypot.f64 (sin.f64 ky) (sin.f64 kx))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 ky) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (sin.f64 ky) (sin.f64 ky)) (*.f64 (sin.f64 kx) (sin.f64 kx)))))) (sin.f64 th)): 29 points increase in error, 2 points decrease in error
    (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 ky) 2)) (*.f64 (sin.f64 kx) (sin.f64 kx))))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 ky) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 kx) 2))))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))) (sin.f64 th)): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.2

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

Alternatives

Alternative 1
Error17.0
Cost52112
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{ky \cdot \sin th}{t_1}\\ \mathbf{if}\;\sin th \leq -0.04:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{elif}\;\sin th \leq 0.58:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin th \leq 0.7:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 2
Error20.1
Cost52112
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{ky \cdot \sin th}{t_1}\\ \mathbf{if}\;\sin ky \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{t_1} \cdot th\\ \mathbf{elif}\;\sin ky \leq -5 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sin ky}{t_1 \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;\sin ky \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 3
Error24.8
Cost39048
\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 4
Error25.0
Cost39048
\[\begin{array}{l} \mathbf{if}\;\sin th \leq -0.04:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\sin th \leq 0.2:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 5
Error0.3
Cost32384
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
Alternative 6
Error37.2
Cost26052
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 7
Error37.2
Cost26052
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sin ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 8
Error37.2
Cost19848
\[\begin{array}{l} \mathbf{if}\;ky \leq -420:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq -3.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{ky \cdot th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{elif}\;ky \leq 3.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 9
Error42.9
Cost19652
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-235}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 10
Error38.1
Cost19652
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-95}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 11
Error38.1
Cost19652
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 5 \cdot 10^{-95}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{\sin th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 12
Error38.1
Cost13384
\[\begin{array}{l} \mathbf{if}\;ky \leq -54:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 10^{-94}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 13
Error43.3
Cost7112
\[\begin{array}{l} \mathbf{if}\;ky \leq -52:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.35 \cdot 10^{-232}:\\ \;\;\;\;th \cdot \left(ky \cdot \frac{1}{\sin kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 14
Error43.4
Cost6984
\[\begin{array}{l} \mathbf{if}\;ky \leq -54:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.85 \cdot 10^{-232}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 15
Error43.4
Cost6984
\[\begin{array}{l} \mathbf{if}\;ky \leq -48:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.85 \cdot 10^{-232}:\\ \;\;\;\;\frac{ky}{\frac{\sin kx}{th}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 16
Error43.3
Cost6984
\[\begin{array}{l} \mathbf{if}\;ky \leq -54:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 1.85 \cdot 10^{-232}:\\ \;\;\;\;\frac{th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 17
Error44.2
Cost6728
\[\begin{array}{l} \mathbf{if}\;ky \leq -54:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 9.2 \cdot 10^{-234}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 18
Error50.5
Cost584
\[\begin{array}{l} \mathbf{if}\;ky \leq -0.003:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2.85 \cdot 10^{-232}:\\ \;\;\;\;ky \cdot \frac{th}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
Alternative 19
Error50.5
Cost584
\[\begin{array}{l} \mathbf{if}\;ky \leq -2.35:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 1.85 \cdot 10^{-232}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
Alternative 20
Error55.0
Cost64
\[th \]

Error

Reproduce

herbie shell --seed 2022331 
(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)))