Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 14.1 → 0.1

Time: 9.1s

Precision: binary64

Cost: 13120

Math

\[\frac{\sin x \cdot \sinh y}{x} \]

↓

\[\frac{\sin x}{x} \cdot \sinh y \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

↓

(FPCore (x y) :precision binary64 (* (/ (sin x) x) (sinh y)))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

↓

double code(double x, double y) {
	return (sin(x) / x) * sinh(y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) / x) * sinh(y)
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

↓

public static double code(double x, double y) {
	return (Math.sin(x) / x) * Math.sinh(y);
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

↓

def code(x, y):
	return (math.sin(x) / x) * math.sinh(y)

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

↓

function code(x, y)
	return Float64(Float64(sin(x) / x) * sinh(y))
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

↓

function tmp = code(x, y)
	tmp = (sin(x) / x) * sinh(y);
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

↓

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision]

Target

Original	14.1
Target	0.3
Herbie	0.1

\[\sin x \cdot \frac{\sinh y}{x} \]

Derivation

1. Initial program 14.1

   \[\frac{\sin x \cdot \sinh y}{x} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
   Proof

   [Start] 14.1	\[ \frac{\sin x \cdot \sinh y}{x} \]
   associate-*l/ [<=] 0.1	\[ \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

3. Final simplification 0.1

   \[\leadsto \frac{\sin x}{x} \cdot \sinh y \]

Alternatives

Alternative 1
Error	0.7
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0002 \lor \neg \left(x \leq 5 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.7
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-5} \lor \neg \left(x \leq 5 \cdot 10^{-23}\right):\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3
Error	0.7
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-5} \lor \neg \left(x \leq 5 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4
Error	0.7
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \left(\sin x \cdot \frac{1}{x}\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-89}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \]

Alternative 5
Error	30.2
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq 1650000000000:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\right) + -1\\ \end{array} \]

Alternative 6
Error	30.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq 1650000000000:\\ \;\;\;\;\frac{1}{\frac{1}{y} + y \cdot -0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)\right) + -1\\ \end{array} \]

Alternative 7
Error	31.3
Cost	576

\[\frac{1}{\frac{1}{y} + y \cdot -0.16666666666666666} \]

Alternative 8
Error	31.4
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022364 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))