Linear.Quaternion:$csinh from linear-1.19.1.3

Average Error: 0.1 → 0.2

Time: 8.3s

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	0.1
Target	0.1
Herbie	0.2

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\sin y \cdot \frac{\cosh x}{y}} \]
   Proof
   (*.f64 (sin.f64 y) (/.f64 (cosh.f64 x) y)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sin.f64 y) (cosh.f64 x)) y)): 8 points increase in error, 43 points decrease in error
   (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (sin.f64 y) y) (cosh.f64 x))): 1 points increase in error, 1 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 29.2

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

4. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   Proof
   (/.f64 (sin.f64 y) (/.f64 y (cosh.f64 x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sin.f64 y) (cosh.f64 x)) y)): 2 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (sin.f64 y) (/.f64 (cosh.f64 x) y))): 43 points increase in error, 8 points decrease in error
   (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (sin.f64 y) (/.f64 (cosh.f64 x) y)) 0)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (sin.f64 y) (/.f64 (cosh.f64 x) y)) (Rewrite<= metadata-eval (-.f64 1 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 (sin.f64 y) (/.f64 (cosh.f64 x) y)) 1) 1)): 133 points increase in error, 20 points decrease in error
   (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (sin.f64 y) (/.f64 (cosh.f64 x) y)))) 1): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.1
Cost	13120

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

Alternative 2
Error	1.0
Cost	7104

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

Alternative 3
Error	1.0
Cost	7104

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

Alternative 4
Error	1.2
Cost	6592

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

Alternative 5
Error	30.9
Cost	6464

\[\cosh x \]

Alternative 6
Error	31.3
Cost	448

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

Alternative 7
Error	31.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022338 
(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)))