Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.6 → 0.2

Time: 10.3s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+61} \lor \neg \left(z \leq 8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.2e+61) (not (<= z 8e-26)))
   (/ (/ x z) (/ y (sin y)))
   (/ x (/ z (/ (sin y) y)))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.2e+61) || !(z <= 8e-26)) {
		tmp = (x / z) / (y / sin(y));
	} else {
		tmp = x / (z / (sin(y) / y));
	}
	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) :: tmp
    if ((z <= (-1.2d+61)) .or. (.not. (z <= 8d-26))) then
        tmp = (x / z) / (y / sin(y))
    else
        tmp = x / (z / (sin(y) / y))
    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 tmp;
	if ((z <= -1.2e+61) || !(z <= 8e-26)) {
		tmp = (x / z) / (y / Math.sin(y));
	} else {
		tmp = x / (z / (Math.sin(y) / y));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (z <= -1.2e+61) or not (z <= 8e-26):
		tmp = (x / z) / (y / math.sin(y))
	else:
		tmp = x / (z / (math.sin(y) / y))
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.2e+61) || !(z <= 8e-26))
		tmp = Float64(Float64(x / z) / Float64(y / sin(y)));
	else
		tmp = Float64(x / Float64(z / Float64(sin(y) / y)));
	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)
	tmp = 0.0;
	if ((z <= -1.2e+61) || ~((z <= 8e-26)))
		tmp = (x / z) / (y / sin(y));
	else
		tmp = x / (z / (sin(y) / y));
	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_] := If[Or[LessEqual[z, -1.2e+61], N[Not[LessEqual[z, 8e-26]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.6
Target	0.3
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 2 regimes

2. if z < -1.1999999999999999e61 or 8.0000000000000003e-26 < z

   1. Initial program 0.1

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

   2. Simplified 5.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      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)): 0 points increase in error, 2 points decrease in error

   3. Applied egg-rr 2.6

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

   4. Applied egg-rr 0.1

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

   if -1.1999999999999999e61 < z < 8.0000000000000003e-26

   1. Initial program 5.2

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      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)): 0 points increase in error, 2 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+61} \lor \neg \left(z \leq 8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	20680

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-114}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	3.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-8} \lor \neg \left(y \leq 4.5 \cdot 10^{-58}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Error	1.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+143} \lor \neg \left(z \leq 1.85 \cdot 10^{+53}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array} \]

Alternative 4
Error	1.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+59} \lor \neg \left(z \leq 3.8 \cdot 10^{+38}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 5
Error	1.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+145} \lor \neg \left(z \leq 2.6 \cdot 10^{+53}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternative 6
Error	2.8
Cost	6848

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

Alternative 7
Error	23.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 34000000000\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}}\\ \end{array} \]

Alternative 8
Error	23.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+49} \lor \neg \left(y \leq 1.45 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 9
Error	22.9
Cost	704

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

Alternative 10
Error	28.8
Cost	192

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

Reproduce

herbie shell --seed 2022343 
(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))