Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Accuracy: 38.6% → 50.9%

Time: 13.4s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (sin(x) / x) * 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] * y), $MachinePrecision]

Target

Original	38.6%
Target	49.4%
Herbie	50.9%

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

Derivation

1. Initial program 37.6%

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

2. Simplified 53.5%

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]
   Step-by-step derivation

   [Start] 37.6	\[ \frac{\sin x \cdot \sinh y}{x} \]
   associate-*l/ [<=] 53.5	\[ \color{blue}{\frac{\sin x}{x} \cdot \sinh y} \]

3. Taylor expanded in y around 0 54.8%

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

4. Final simplification 54.8%

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

Alternatives

Alternative 1
Accuracy	32.5%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 5.9 \cdot 10^{+26}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]

Alternative 2
Accuracy	50.0%
Cost	6720

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

Alternative 3
Accuracy	27.8%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023157 
(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))