Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.8 → 0.6

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq -3.597189445155209 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 1.3368765705641742 \cdot 10^{-120}:\\ \;\;\;\;\frac{x}{z} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]

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 -3.597189445155209e+89)
     (/ (/ (* x (sin y)) y) z)
     (if (<= x 1.3368765705641742e-120) (* (/ 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 <= -3.597189445155209e+89) {
		tmp = ((x * sin(y)) / y) / z;
	} else if (x <= 1.3368765705641742e-120) {
		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 <= (-3.597189445155209d+89)) then
        tmp = ((x * sin(y)) / y) / z
    else if (x <= 1.3368765705641742d-120) 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 <= -3.597189445155209e+89) {
		tmp = ((x * Math.sin(y)) / y) / z;
	} else if (x <= 1.3368765705641742e-120) {
		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 <= -3.597189445155209e+89:
		tmp = ((x * math.sin(y)) / y) / z
	elif x <= 1.3368765705641742e-120:
		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 <= -3.597189445155209e+89)
		tmp = Float64(Float64(Float64(x * sin(y)) / y) / z);
	elseif (x <= 1.3368765705641742e-120)
		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 <= -3.597189445155209e+89)
		tmp = ((x * sin(y)) / y) / z;
	elseif (x <= 1.3368765705641742e-120)
		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, -3.597189445155209e+89], N[(N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 1.3368765705641742e-120], N[(N[(x / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	2.8
Target	0.2
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if x < -3.59718944515520873e89

   1. Initial program 0.2

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

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \frac{x}{\frac{y}{\sin y}}} \]

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.3

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

   if -3.59718944515520873e89 < x < 1.3368765705641742e-120

   1. Initial program 4.8

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

   2. Applied egg-rr 0.4

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

   if 1.3368765705641742e-120 < x

   1. Initial program 0.9

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

   2. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \frac{x}{\frac{y}{\sin y}}} \]

   3. Applied egg-rr 0.9

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.597189445155209 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 1.3368765705641742 \cdot 10^{-120}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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))