Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 3.0 → 2.3

Time: 8.6s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-210}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e-210) (/ (/ x (/ y (sin y))) z) (/ 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 <= -2.3e-210) {
		tmp = (x / (y / sin(y))) / z;
	} 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 <= (-2.3d-210)) then
        tmp = (x / (y / sin(y))) / z
    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 <= -2.3e-210) {
		tmp = (x / (y / Math.sin(y))) / z;
	} 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 <= -2.3e-210:
		tmp = (x / (y / math.sin(y))) / z
	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 <= -2.3e-210)
		tmp = Float64(Float64(x / Float64(y / sin(y))) / z);
	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 <= -2.3e-210)
		tmp = (x / (y / sin(y))) / z;
	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[LessEqual[z, -2.3e-210], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z / N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	3.0
Target	0.3
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if z < -2.3e-210

   1. Initial program 2.0

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

   2. Simplified 2.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{y}{\sin y}}}{z}} \]
      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, 12 points decrease in error
      (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 x (/.f64 (sin.f64 y) y))) z): 10 points increase in error, 37 points decrease in error

   if -2.3e-210 < z

   1. Initial program 3.7

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

   2. Simplified 2.6

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      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)): 23 points increase in error, 21 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-210}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.8
Cost	7112

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3
Error	1.5
Cost	6980

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \end{array} \]

Alternative 4
Error	2.8
Cost	6848

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

Alternative 5
Error	22.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+14}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 690:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y \cdot y\right) \cdot -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z} \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	840

\[\begin{array}{l} t_0 := 6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{if}\;y \leq -2400000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	22.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -850000:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{z} \cdot \frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 8
Error	23.3
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z \cdot y}\\ \mathbf{if}\;y \leq -0.0185:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq 1000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]

Alternative 10
Error	23.2
Cost	712

\[\begin{array}{l} t_0 := \left(\frac{x}{z} + 1\right) + -1\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 550:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	22.7
Cost	704

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

Alternative 12
Error	28.3
Cost	192

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

Reproduce

herbie shell --seed 2022331 
(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))