Linear.Quaternion:$csinh from linear-1.19.1.3

Average Error: 0.1 → 0.7

Time: 21.1s

Precision: binary64

Cost: 46720

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_0 + \left(0.5 \cdot \left({x}^{2} \cdot t_0\right) + \left(0.041666666666666664 \cdot \left({x}^{4} \cdot t_0\right) + 0.001388888888888889 \cdot \left({x}^{6} \cdot t_0\right)\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (+
    t_0
    (+
     (* 0.5 (* (pow x 2.0) t_0))
     (+
      (* 0.041666666666666664 (* (pow x 4.0) t_0))
      (* 0.001388888888888889 (* (pow x 6.0) t_0)))))))

C

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

↓

double code(double x, double y) {
	double t_0 = sin(y) / y;
	return t_0 + ((0.5 * (pow(x, 2.0) * t_0)) + ((0.041666666666666664 * (pow(x, 4.0) * t_0)) + (0.001388888888888889 * (pow(x, 6.0) * t_0))));
}

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
    real(8) :: t_0
    t_0 = sin(y) / y
    code = t_0 + ((0.5d0 * ((x ** 2.0d0) * t_0)) + ((0.041666666666666664d0 * ((x ** 4.0d0) * t_0)) + (0.001388888888888889d0 * ((x ** 6.0d0) * t_0))))
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) {
	double t_0 = Math.sin(y) / y;
	return t_0 + ((0.5 * (Math.pow(x, 2.0) * t_0)) + ((0.041666666666666664 * (Math.pow(x, 4.0) * t_0)) + (0.001388888888888889 * (Math.pow(x, 6.0) * t_0))));
}

Python

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

↓

def code(x, y):
	t_0 = math.sin(y) / y
	return t_0 + ((0.5 * (math.pow(x, 2.0) * t_0)) + ((0.041666666666666664 * (math.pow(x, 4.0) * t_0)) + (0.001388888888888889 * (math.pow(x, 6.0) * t_0))))

Julia

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

↓

function code(x, y)
	t_0 = Float64(sin(y) / y)
	return Float64(t_0 + Float64(Float64(0.5 * Float64((x ^ 2.0) * t_0)) + Float64(Float64(0.041666666666666664 * Float64((x ^ 4.0) * t_0)) + Float64(0.001388888888888889 * Float64((x ^ 6.0) * t_0)))))
end

MATLAB

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

↓

function tmp = code(x, y)
	t_0 = sin(y) / y;
	tmp = t_0 + ((0.5 * ((x ^ 2.0) * t_0)) + ((0.041666666666666664 * ((x ^ 4.0) * t_0)) + (0.001388888888888889 * ((x ^ 6.0) * t_0))));
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(t$95$0 + N[(N[(0.5 * N[(N[Power[x, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(N[Power[x, 4.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[(N[Power[x, 6.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.1
Target	0.1
Herbie	0.7

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

Derivation

1. Initial program 0.1

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

2. Taylor expanded in x around 0 0.7

   \[\leadsto \color{blue}{\frac{\sin y}{y} + \left(0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y} + \left(0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y} + 0.001388888888888889 \cdot \frac{\sin y \cdot {x}^{6}}{y}\right)\right)} \]

3. Simplified 0.7

   \[\leadsto \color{blue}{\frac{\sin y}{y} + \left(0.5 \cdot \left({x}^{2} \cdot \frac{\sin y}{y}\right) + \left(0.041666666666666664 \cdot \left({x}^{4} \cdot \frac{\sin y}{y}\right) + 0.001388888888888889 \cdot \left({x}^{6} \cdot \frac{\sin y}{y}\right)\right)\right)} \]
   Proof

   [Start] 0.7	\[ \frac{\sin y}{y} + \left(0.5 \cdot \frac{\sin y \cdot {x}^{2}}{y} + \left(0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y} + 0.001388888888888889 \cdot \frac{\sin y \cdot {x}^{6}}{y}\right)\right) \]
   rational.json-simplify-49 [=>] 0.7	\[ \frac{\sin y}{y} + \left(0.5 \cdot \color{blue}{\left({x}^{2} \cdot \frac{\sin y}{y}\right)} + \left(0.041666666666666664 \cdot \frac{\sin y \cdot {x}^{4}}{y} + 0.001388888888888889 \cdot \frac{\sin y \cdot {x}^{6}}{y}\right)\right) \]
   rational.json-simplify-49 [=>] 0.7	\[ \frac{\sin y}{y} + \left(0.5 \cdot \left({x}^{2} \cdot \frac{\sin y}{y}\right) + \left(0.041666666666666664 \cdot \color{blue}{\left({x}^{4} \cdot \frac{\sin y}{y}\right)} + 0.001388888888888889 \cdot \frac{\sin y \cdot {x}^{6}}{y}\right)\right) \]
   rational.json-simplify-49 [=>] 0.7	\[ \frac{\sin y}{y} + \left(0.5 \cdot \left({x}^{2} \cdot \frac{\sin y}{y}\right) + \left(0.041666666666666664 \cdot \left({x}^{4} \cdot \frac{\sin y}{y}\right) + 0.001388888888888889 \cdot \color{blue}{\left({x}^{6} \cdot \frac{\sin y}{y}\right)}\right)\right) \]

4. Final simplification 0.7

   \[\leadsto \frac{\sin y}{y} + \left(0.5 \cdot \left({x}^{2} \cdot \frac{\sin y}{y}\right) + \left(0.041666666666666664 \cdot \left({x}^{4} \cdot \frac{\sin y}{y}\right) + 0.001388888888888889 \cdot \left({x}^{6} \cdot \frac{\sin y}{y}\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	13120

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

Alternative 2
Error	31.5
Cost	6592

\[\cosh x \cdot 1 \]

Alternative 3
Error	1.4
Cost	6592

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

Alternative 4
Error	32.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023075 
(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)))