Linear.Quaternion:$csinh from linear-1.19.1.3

Average Accuracy: 51.0% → 51.9%

Time: 9.5s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (+ (/ (sin y) y) (* (/ x (/ y (sin y))) (* x 0.5))))

C

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

↓

double code(double x, double y) {
	return (sin(y) / y) + ((x / (y / sin(y))) * (x * 0.5));
}

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) + ((x / (y / sin(y))) * (x * 0.5d0))
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) + ((x / (y / Math.sin(y))) * (x * 0.5));
}

Python

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

↓

def code(x, y):
	return (math.sin(y) / y) + ((x / (y / math.sin(y))) * (x * 0.5))

Julia

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

↓

function code(x, y)
	return Float64(Float64(sin(y) / y) + Float64(Float64(x / Float64(y / sin(y))) * Float64(x * 0.5)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (sin(y) / y) + ((x / (y / sin(y))) * (x * 0.5));
end

Wolfram

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

↓

code[x_, y_] := N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] + N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	51.0%
Target	51.0%
Herbie	51.9%

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

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in x around 0 49.3%

   \[\leadsto \color{blue}{\frac{\sin y}{y} + \left(0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y} + 0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y}\right)} \]

3. Simplified 49.3%

   \[\leadsto \color{blue}{\frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \left(x \cdot x\right), 0.041666666666666664 \cdot \left(\frac{\sin y}{y} \cdot {x}^{4}\right)\right)} \]
   Step-by-step derivation

   [Start] 49.3	\[ \frac{\sin y}{y} + \left(0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y} + 0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y}\right) \]
   fma-def [=>] 49.3	\[ \frac{\sin y}{y} + \color{blue}{\mathsf{fma}\left(0.5, \frac{\sin y \cdot {x}^{2}}{y}, 0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y}\right)} \]
   associate-/l* [=>] 49.2	\[ \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \color{blue}{\frac{\sin y}{\frac{y}{{x}^{2}}}}, 0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y}\right) \]
   associate-/r/ [=>] 49.3	\[ \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \color{blue}{\frac{\sin y}{y} \cdot {x}^{2}}, 0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y}\right) \]
   unpow2 [=>] 49.3	\[ \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \color{blue}{\left(x \cdot x\right)}, 0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y}\right) \]
   associate-/l* [=>] 49.1	\[ \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \left(x \cdot x\right), 0.041666666666666664 \cdot \color{blue}{\frac{\sin y}{\frac{y}{{x}^{4}}}}\right) \]
   associate-/r/ [=>] 49.3	\[ \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \left(x \cdot x\right), 0.041666666666666664 \cdot \color{blue}{\left(\frac{\sin y}{y} \cdot {x}^{4}\right)}\right) \]

4. Taylor expanded in y around 0 44.9%

   \[\leadsto \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \left(x \cdot x\right), \color{blue}{0.041666666666666664 \cdot {x}^{4}}\right) \]

5. Simplified 44.9%

   \[\leadsto \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \left(x \cdot x\right), \color{blue}{{x}^{4} \cdot 0.041666666666666664}\right) \]
   Step-by-step derivation

   [Start] 44.9	\[ \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \left(x \cdot x\right), 0.041666666666666664 \cdot {x}^{4}\right) \]
   *-commutative [=>] 44.9	\[ \frac{\sin y}{y} + \mathsf{fma}\left(0.5, \frac{\sin y}{y} \cdot \left(x \cdot x\right), \color{blue}{{x}^{4} \cdot 0.041666666666666664}\right) \]

6. Taylor expanded in x around 0 49.9%

   \[\leadsto \frac{\sin y}{y} + \color{blue}{0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y}} \]

7. Simplified 50.3%

   \[\leadsto \frac{\sin y}{y} + \color{blue}{\frac{x}{\frac{y}{\sin y}} \cdot \left(0.5 \cdot x\right)} \]
   Step-by-step derivation

   [Start] 49.9	\[ \frac{\sin y}{y} + 0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y} \]
   associate-*l/ [<=] 49.9	\[ \frac{\sin y}{y} + 0.5 \cdot \color{blue}{\left(\frac{\sin y}{y} \cdot {x}^{2}\right)} \]
   unpow2 [=>] 49.9	\[ \frac{\sin y}{y} + 0.5 \cdot \left(\frac{\sin y}{y} \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
   associate-*r* [=>] 50.3	\[ \frac{\sin y}{y} + 0.5 \cdot \color{blue}{\left(\left(\frac{\sin y}{y} \cdot x\right) \cdot x\right)} \]
   associate-*l* [<=] 50.3	\[ \frac{\sin y}{y} + \color{blue}{\left(0.5 \cdot \left(\frac{\sin y}{y} \cdot x\right)\right) \cdot x} \]
   *-commutative [=>] 50.3	\[ \frac{\sin y}{y} + \color{blue}{\left(\left(\frac{\sin y}{y} \cdot x\right) \cdot 0.5\right)} \cdot x \]
   associate-*l* [=>] 50.3	\[ \frac{\sin y}{y} + \color{blue}{\left(\frac{\sin y}{y} \cdot x\right) \cdot \left(0.5 \cdot x\right)} \]
   *-commutative [=>] 50.3	\[ \frac{\sin y}{y} + \color{blue}{\left(x \cdot \frac{\sin y}{y}\right)} \cdot \left(0.5 \cdot x\right) \]
   associate-*r/ [=>] 50.3	\[ \frac{\sin y}{y} + \color{blue}{\frac{x \cdot \sin y}{y}} \cdot \left(0.5 \cdot x\right) \]
   associate-/l* [=>] 50.3	\[ \frac{\sin y}{y} + \color{blue}{\frac{x}{\frac{y}{\sin y}}} \cdot \left(0.5 \cdot x\right) \]

8. Final simplification 50.3%

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

Alternatives

Alternative 1
Accuracy	51.4%
Cost	6592

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

Alternative 2
Accuracy	26.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023157 
(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)))