Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.6 → 0.5

Time: 1.8min

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{\sin y \cdot x}{y}}{z}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{z} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= x -2e+115)
     (/ (/ (* (sin y) x) y) z)
     (if (<= x 2e+68) (* (/ x z) t_0) (/ (* x t_0) z)))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x <= -2e+115) {
		tmp = ((sin(y) * x) / y) / z;
	} else if (x <= 2e+68) {
		tmp = (x / z) * t_0;
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (sin(y) / y)) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x <= (-2d+115)) then
        tmp = ((sin(y) * x) / y) / z
    else if (x <= 2d+68) then
        tmp = (x / z) * t_0
    else
        tmp = (x * t_0) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x <= -2e+115) {
		tmp = ((Math.sin(y) * x) / y) / z;
	} else if (x <= 2e+68) {
		tmp = (x / z) * t_0;
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x <= -2e+115:
		tmp = ((math.sin(y) * x) / y) / z
	elif x <= 2e+68:
		tmp = (x / z) * t_0
	else:
		tmp = (x * t_0) / z
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x <= -2e+115)
		tmp = Float64(Float64(Float64(sin(y) * x) / y) / z);
	elseif (x <= 2e+68)
		tmp = Float64(Float64(x / z) * t_0);
	else
		tmp = Float64(Float64(x * t_0) / z);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x <= -2e+115)
		tmp = ((sin(y) * x) / y) / z;
	elseif (x <= 2e+68)
		tmp = (x / z) * t_0;
	else
		tmp = (x * t_0) / z;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -2e+115], N[(N[(N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 2e+68], N[(N[(x / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	2.6
Target	0.2
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -2e115

   1. Initial program 0.2

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

   2. Taylor expanded in x around 0 0.3

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

   if -2e115 < x < 1.99999999999999991e68

   1. Initial program 3.6

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

   2. Simplified 0.6

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

   if 1.99999999999999991e68 < x

   1. Initial program 0.2

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	0.3
Cost	20424

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ t_2 := \frac{t_1}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-146}:\\ \;\;\;\;\frac{x}{z} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	0.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x}{z} \cdot t_0\\ \mathbf{if}\;z \leq -8 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{t_0}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	3.2
Cost	6848

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

Alternative 4
Error	22.9
Cost	712

\[\begin{array}{l} t_0 := 3 - \left(3 - \frac{x}{z}\right)\\ \mathbf{if}\;y \leq -11500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	27.9
Cost	192

\[\frac{x}{z} \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))