Linear.Quaternion:$ccos from linear-1.19.1.3

Average Error: 0.0 → 0.0

Time: 8.8s

Precision: binary64

Cost: 13248

Math

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

↓

\[\frac{\sin x}{\frac{1}{\frac{\sinh y}{y}}} \]

FPCore

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

↓

(FPCore (x y) :precision binary64 (/ (sin x) (/ 1.0 (/ (sinh y) y))))

C

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

↓

double code(double x, double y) {
	return sin(x) / (1.0 / (sinh(y) / y));
}

Fortran

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) / (1.0d0 / (sinh(y) / y))
end function

Java

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

↓

public static double code(double x, double y) {
	return Math.sin(x) / (1.0 / (Math.sinh(y) / y));
}

Python

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

↓

def code(x, y):
	return math.sin(x) / (1.0 / (math.sinh(y) / y))

Julia

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

↓

function code(x, y)
	return Float64(sin(x) / Float64(1.0 / Float64(sinh(y) / y)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = sin(x) / (1.0 / (sinh(y) / y));
end

Wolfram

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

↓

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] / N[(1.0 / N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]
   Proof
   (/.f64 (sin.f64 x) (/.f64 y (sinh.f64 y))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) y)): 83 points increase in error, 4 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))): 6 points increase in error, 82 points decrease in error

3. Applied egg-rr 0.2

   \[\leadsto \frac{\sin x}{\color{blue}{\frac{1}{\sinh y} \cdot y}} \]

4. Applied egg-rr 0.0

   \[\leadsto \frac{\sin x}{\color{blue}{\frac{1}{\frac{\sinh y}{y}}}} \]

5. Final simplification 0.0

   \[\leadsto \frac{\sin x}{\frac{1}{\frac{\sinh y}{y}}} \]

Alternatives

Alternative 1
Error	0.0
Cost	13120

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

Alternative 2
Error	0.0
Cost	13120

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

Alternative 3
Error	0.7
Cost	6976

\[\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

Alternative 4
Error	1.0
Cost	6464

\[\sin x \]

Alternative 5
Error	31.8
Cost	576

\[x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

Alternative 6
Error	32.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022337 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))