Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 14.0 → 0.1

Time: 4.8s

Precision: binary64

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]

Error

Bits error versus x

Bits error versus y

Target

Original	14.0
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 14.0

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

2. Taylor expanded in x around inf 43.3

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sin x \cdot e^{y}\right) - 0.5 \cdot \frac{\sin x}{e^{y}}}{x}} \]

3. Simplified 0.3

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

4. Applied div-inv_binary64 0.4

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

5. Applied associate-*l*_binary64 0.2

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

6. Simplified 0.1

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

7. Applied *-commutative_binary64 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022131 
(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))