Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.4 → 0.2

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ t_1 := \frac{y}{\sin y}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{x}{t_1}}{z}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{x}{z}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (/ (sin y) y)) z)) (t_1 (/ y (sin y))))
   (if (<= t_0 -2e-131)
     (/ (/ x t_1) z)
     (if (<= t_0 4e-92) (/ (/ x z) t_1) t_0))))

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 = (x * (sin(y) / y)) / z;
	double t_1 = y / sin(y);
	double tmp;
	if (t_0 <= -2e-131) {
		tmp = (x / t_1) / z;
	} else if (t_0 <= 4e-92) {
		tmp = (x / z) / t_1;
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x * (sin(y) / y)) / z
    t_1 = y / sin(y)
    if (t_0 <= (-2d-131)) then
        tmp = (x / t_1) / z
    else if (t_0 <= 4d-92) then
        tmp = (x / z) / t_1
    else
        tmp = t_0
    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 = (x * (Math.sin(y) / y)) / z;
	double t_1 = y / Math.sin(y);
	double tmp;
	if (t_0 <= -2e-131) {
		tmp = (x / t_1) / z;
	} else if (t_0 <= 4e-92) {
		tmp = (x / z) / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x * (math.sin(y) / y)) / z
	t_1 = y / math.sin(y)
	tmp = 0
	if t_0 <= -2e-131:
		tmp = (x / t_1) / z
	elif t_0 <= 4e-92:
		tmp = (x / z) / t_1
	else:
		tmp = t_0
	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(Float64(x * Float64(sin(y) / y)) / z)
	t_1 = Float64(y / sin(y))
	tmp = 0.0
	if (t_0 <= -2e-131)
		tmp = Float64(Float64(x / t_1) / z);
	elseif (t_0 <= 4e-92)
		tmp = Float64(Float64(x / z) / t_1);
	else
		tmp = t_0;
	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 = (x * (sin(y) / y)) / z;
	t_1 = y / sin(y);
	tmp = 0.0;
	if (t_0 <= -2e-131)
		tmp = (x / t_1) / z;
	elseif (t_0 <= 4e-92)
		tmp = (x / z) / t_1;
	else
		tmp = t_0;
	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[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-131], N[(N[(x / t$95$1), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 4e-92], N[(N[(x / z), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$0]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.4
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2e-131

   1. Initial program 0.3

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

   2. Applied egg-rr 0.4

      \[\leadsto \frac{\color{blue}{{\left(\frac{x}{\frac{y}{\sin y}}\right)}^{1}}}{z} \]

   if -2e-131 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 3.99999999999999995e-92

   1. Initial program 4.7

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

   2. Applied egg-rr 4.8

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

   3. Applied egg-rr 0.1

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

   if 3.99999999999999995e-92 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 0.2

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

   2. Applied egg-rr 0.4

      \[\leadsto \frac{x \cdot \color{blue}{\left(\sin y \cdot \frac{1}{y}\right)}}{z} \]

   3. Applied egg-rr 0.2

      \[\leadsto \frac{\color{blue}{{\left(x \cdot \frac{\sin y}{y}\right)}^{1}}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))