Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.9 → 0.4

Time: 10.2s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

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 -2.1e+90)
     (/ (* x t_0) z)
     (if (<= x 2e+30) (/ x (/ z t_0)) (/ (/ x (/ y (sin y))) 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 <= -2.1e+90) {
		tmp = (x * t_0) / z;
	} else if (x <= 2e+30) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / (y / sin(y))) / 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 <= (-2.1d+90)) then
        tmp = (x * t_0) / z
    else if (x <= 2d+30) then
        tmp = x / (z / t_0)
    else
        tmp = (x / (y / sin(y))) / 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 <= -2.1e+90) {
		tmp = (x * t_0) / z;
	} else if (x <= 2e+30) {
		tmp = x / (z / t_0);
	} else {
		tmp = (x / (y / Math.sin(y))) / 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 <= -2.1e+90:
		tmp = (x * t_0) / z
	elif x <= 2e+30:
		tmp = x / (z / t_0)
	else:
		tmp = (x / (y / math.sin(y))) / 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 <= -2.1e+90)
		tmp = Float64(Float64(x * t_0) / z);
	elseif (x <= 2e+30)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(x / Float64(y / sin(y))) / 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 <= -2.1e+90)
		tmp = (x * t_0) / z;
	elseif (x <= 2e+30)
		tmp = x / (z / t_0);
	else
		tmp = (x / (y / sin(y))) / 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, -2.1e+90], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 2e+30], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	2.9
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999981e90

   1. Initial program 0.2

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

   if -2.09999999999999981e90 < x < 2e30

   1. Initial program 4.3

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      Proof
      (/.f64 x (/.f64 z (/.f64 (sin.f64 y) y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)): 30 points increase in error, 18 points decrease in error

   if 2e30 < x

   1. Initial program 0.2

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{y}{\sin y}}}{z}} \]
      Proof
      (/.f64 (/.f64 x (/.f64 y (sin.f64 y))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (sin.f64 y)) y)) z): 40 points increase in error, 11 points decrease in error
      (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 (sin.f64 y) y))) z): 16 points increase in error, 41 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.2
Cost	7112

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{if}\;y \leq -0.00029:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.2
Cost	7112

\[\begin{array}{l} t_0 := \sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{if}\;y \leq -0.0005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.00032:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0002:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4
Error	1.0
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+185}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	3.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\sin y}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 6
Error	3.2
Cost	6848

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

Alternative 7
Error	23.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{6}{\frac{y}{\frac{x}{y}}}}{z}\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 8
Error	23.1
Cost	840

\[\begin{array}{l} t_0 := 6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.1
Cost	840

\[\begin{array}{l} t_0 := 6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	23.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 11
Error	23.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;\frac{6 \cdot \frac{\frac{x}{y}}{y}}{z}\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 12
Error	23.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;\frac{\frac{6}{\frac{y}{\frac{x}{y}}}}{z}\\ \mathbf{elif}\;y \leq 0.042:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 13
Error	23.7
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{y \cdot z}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.00019:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	23.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 15
Error	23.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{\frac{y \cdot z}{x}}\\ \mathbf{elif}\;y \leq 0.0095:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 16
Error	23.0
Cost	704

\[\frac{\frac{x}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}}{z} \]

Alternative 17
Error	28.9
Cost	192

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

Reproduce

herbie shell --seed 2022338 
(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))