Linear.Quaternion:$csinh from linear-1.19.1.3

Average Error: 0.2 → 0.2

Time: 10.9s

Precision: binary64

Cost: 13120

Math

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

↓

\[\frac{\sin y}{\frac{y}{\cosh x}} \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (/ (sin y) (/ y (cosh x))))

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{\cosh x}{\frac{y}{\sin y}}} \]

3. Applied egg-rr 29.7

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin y \cdot \frac{\cosh x}{y}\right)} - 1} \]

4. Simplified 0.2

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

   [Start] 29.7	\[ e^{\mathsf{log1p}\left(\sin y \cdot \frac{\cosh x}{y}\right)} - 1 \]
   expm1-def [=>] 0.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \frac{\cosh x}{y}\right)\right)} \]
   expm1-log1p [=>] 0.3	\[ \color{blue}{\sin y \cdot \frac{\cosh x}{y}} \]
   associate-*r/ [=>] 0.2	\[ \color{blue}{\frac{\sin y \cdot \cosh x}{y}} \]
   associate-/l* [=>] 0.2	\[ \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

5. Final simplification 0.2

   \[\leadsto \frac{\sin y}{\frac{y}{\cosh x}} \]

Alternatives

Alternative 1
Error	0.2
Cost	13120

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

Alternative 2
Error	1.0
Cost	7104

\[\frac{1 + x \cdot \left(x \cdot 0.5\right)}{\frac{y}{\sin y}} \]

Alternative 3
Error	1.3
Cost	6592

\[\frac{\sin y}{y} \]

Alternative 4
Error	30.9
Cost	1216

\[\frac{\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}}{\frac{1}{y} + y \cdot 0.16666666666666666} \]

Alternative 5
Error	31.7
Cost	448

\[1 + x \cdot \left(x \cdot 0.5\right) \]

Alternative 6
Error	31.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

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

  (* (cosh x) (/ (sin y) y)))