Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.9 → 1.7

Time: 23.1s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z} \cdot x\\ \end{array} \]

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 (<= z -5.2e-104) (/ (* x t_0) z) (* (/ t_0 z) x))))

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 (z <= -5.2e-104) {
		tmp = (x * t_0) / z;
	} else {
		tmp = (t_0 / z) * x;
	}
	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 (z <= (-5.2d-104)) then
        tmp = (x * t_0) / z
    else
        tmp = (t_0 / z) * x
    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 (z <= -5.2e-104) {
		tmp = (x * t_0) / z;
	} else {
		tmp = (t_0 / z) * x;
	}
	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 z <= -5.2e-104:
		tmp = (x * t_0) / z
	else:
		tmp = (t_0 / z) * x
	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 (z <= -5.2e-104)
		tmp = Float64(Float64(x * t_0) / z);
	else
		tmp = Float64(Float64(t_0 / z) * x);
	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 (z <= -5.2e-104)
		tmp = (x * t_0) / z;
	else
		tmp = (t_0 / z) * x;
	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[z, -5.2e-104], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$0 / z), $MachinePrecision] * x), $MachinePrecision]]]

Target

Original	2.9
Target	0.3
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000005e-104

   1. Initial program 0.7

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

   if -5.20000000000000005e-104 < z

   1. Initial program 4.2

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

   2. Applied egg-rr 2.3

      \[\leadsto \color{blue}{\frac{\frac{\sin y}{y}}{z} \cdot x} \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	2.9
Cost	7112

\[\begin{array}{l} t_0 := \sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.00029:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot -0.16666666666666666\right) \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 3
Error	3.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4
Error	3.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5
Error	2.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-27}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y}}{z} \cdot x\\ \end{array} \]

Alternative 6
Error	23.3
Cost	1744

\[\begin{array}{l} t_0 := \frac{y \cdot x}{z}\\ t_1 := \frac{y \cdot x}{z \cdot y}\\ \mathbf{if}\;y \leq -195000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot -0.16666666666666666\right) \cdot y\right)}{z}\\ \mathbf{elif}\;t_0 \ne 0:\\ \;\;\;\;\frac{1}{\begin{array}{l} \mathbf{if}\;\frac{y}{x} \ne 0:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{\frac{1}{z}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0}\\ \end{array}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	23.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1650000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z \cdot y}\\ \end{array}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{1 + \left(y \cdot y\right) \cdot -0.16666666666666666}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 8
Error	23.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -64000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z \cdot y}\\ \end{array}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y \cdot -0.16666666666666666\right) \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 9
Error	23.7
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot y}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	23.7
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;x \ne 0:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z \cdot y}\\ \end{array}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]

Alternative 12
Error	28.7
Cost	192

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

Reproduce

herbie shell --seed 2023010 
(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))