Average Error: 4.3 → 0.2
Time: 23.4s
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 4.3

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

  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)): 18 points increase in error, 3 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
Error0.3
Cost32384
\[\sin ky \cdot \frac{\sin th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \]
Alternative 2
Error15.4
Cost26632
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{ky \cdot \sin th}{t_1}\\ \mathbf{if}\;th \leq -0.41:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 32:\\ \;\;\;\;\frac{\sin ky}{t_1 \cdot \left(\frac{1}{th} + th \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 3
Error25.1
Cost26380
\[\begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{if}\;th \leq -1.45 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;th \leq -0.82:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 4.5 \cdot 10^{+33}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error15.5
Cost26248
\[\begin{array}{l} t_1 := \mathsf{hypot}\left(\sin ky, \sin kx\right)\\ t_2 := \frac{ky \cdot \sin th}{t_1}\\ \mathbf{if}\;th \leq -0.41:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 32:\\ \;\;\;\;\sin ky \cdot \frac{th}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error34.5
Cost26184
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{{\sin th}^{2}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 6
Error42.3
Cost19652
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq 4 \cdot 10^{-178}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 7
Error37.9
Cost13384
\[\begin{array}{l} \mathbf{if}\;ky \leq -7000000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 2.35 \cdot 10^{-66}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 8
Error42.8
Cost6984
\[\begin{array}{l} \mathbf{if}\;ky \leq -740000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 9
Error43.8
Cost6728
\[\begin{array}{l} \mathbf{if}\;ky \leq -7000000000:\\ \;\;\;\;\sin th\\ \mathbf{elif}\;ky \leq 6.4 \cdot 10^{-174}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
Alternative 10
Error49.9
Cost584
\[\begin{array}{l} \mathbf{if}\;ky \leq -7000000000:\\ \;\;\;\;th\\ \mathbf{elif}\;ky \leq 2 \cdot 10^{-68}:\\ \;\;\;\;th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
Alternative 11
Error55.2
Cost64
\[th \]

Error

Reproduce

herbie shell --seed 2022329 
(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)))