Linear.Quaternion:$ccosh from linear-1.19.1.3

Average Error: 14.2 → 0.3

Time: 1.2min

Precision: binary64

Cost: 13384

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-60}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= x -2.1e-8) t_0 (if (<= x 4.1e-60) (sinh y) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (x <= -2.1e-8) {
		tmp = t_0;
	} else if (x <= 4.1e-60) {
		tmp = sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) * sinh(y)) / x
    if (x <= (-2.1d-8)) then
        tmp = t_0
    else if (x <= 4.1d-60) then
        tmp = sinh(y)
    else
        tmp = t_0
    end if
    code = tmp
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) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if (x <= -2.1e-8) {
		tmp = t_0;
	} else if (x <= 4.1e-60) {
		tmp = Math.sinh(y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if x <= -2.1e-8:
		tmp = t_0
	elif x <= 4.1e-60:
		tmp = math.sinh(y)
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (x <= -2.1e-8)
		tmp = t_0;
	elseif (x <= 4.1e-60)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (x <= -2.1e-8)
		tmp = t_0;
	elseif (x <= 4.1e-60)
		tmp = sinh(y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -2.1e-8], t$95$0, If[LessEqual[x, 4.1e-60], N[Sinh[y], $MachinePrecision], t$95$0]]]

Target

Original	14.2
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999994e-8 or 4.10000000000000013e-60 < x

   1. Initial program 0.6

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

   if -2.09999999999999994e-8 < x < 4.10000000000000013e-60

   1. Initial program 31.0

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

   2. Taylor expanded in x around 0 31.0

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

   3. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\sinh y + 0} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\sinh y} \]
      Proof

      [Start] 0.0	\[ \sinh y + 0 \]
      rational_best-simplify-3 [<=] 0.0	\[ \color{blue}{0 + \sinh y} \]
      rational_best-simplify-6 [=>] 0.0	\[ \color{blue}{\sinh y} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x \cdot \sinh y}{x}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-60}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \sinh y}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13568

\[\frac{0}{\frac{1}{x}} - \sin x \cdot \frac{-\sinh y}{x} \]

Alternative 2
Error	0.8
Cost	6984

\[\begin{array}{l} t_0 := \frac{\sin x \cdot y}{x}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-9}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.6
Cost	6728

\[\begin{array}{l} t_0 := \left(y \cdot 0.5 - \left(-1 - y \cdot 0.5\right)\right) + -1\\ \mathbf{if}\;x \leq -550:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+34}:\\ \;\;\;\;\sinh y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	18.2
Cost	320

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

Alternative 5
Error	31.0
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023099 
(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))