Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.5 → 0.7

Time: 11.8s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{+109}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

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)))
   (if (<= t_0 -5e-199)
     t_0
     (if (<= t_0 1e+109)
       (/ (/ x z) (/ y (sin y)))
       (/ x (* y (/ z (sin 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 t_0 = (x * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-199) {
		tmp = t_0;
	} else if (t_0 <= 1e+109) {
		tmp = (x / z) / (y / sin(y));
	} else {
		tmp = x / (y * (z / sin(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) :: t_0
    real(8) :: tmp
    t_0 = (x * (sin(y) / y)) / z
    if (t_0 <= (-5d-199)) then
        tmp = t_0
    else if (t_0 <= 1d+109) then
        tmp = (x / z) / (y / sin(y))
    else
        tmp = x / (y * (z / sin(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 t_0 = (x * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-199) {
		tmp = t_0;
	} else if (t_0 <= 1e+109) {
		tmp = (x / z) / (y / Math.sin(y));
	} else {
		tmp = x / (y * (z / Math.sin(y)));
	}
	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
	tmp = 0
	if t_0 <= -5e-199:
		tmp = t_0
	elif t_0 <= 1e+109:
		tmp = (x / z) / (y / math.sin(y))
	else:
		tmp = x / (y * (z / math.sin(y)))
	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)
	tmp = 0.0
	if (t_0 <= -5e-199)
		tmp = t_0;
	elseif (t_0 <= 1e+109)
		tmp = Float64(Float64(x / z) / Float64(y / sin(y)));
	else
		tmp = Float64(x / Float64(y * Float64(z / sin(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)
	t_0 = (x * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -5e-199)
		tmp = t_0;
	elseif (t_0 <= 1e+109)
		tmp = (x / z) / (y / sin(y));
	else
		tmp = x / (y * (z / sin(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_] := Block[{t$95$0 = N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-199], t$95$0, If[LessEqual[t$95$0, 1e+109], N[(N[(x / z), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	2.5
Target	0.3
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -4.9999999999999996e-199

   1. Initial program 0.3

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

   if -4.9999999999999996e-199 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.99999999999999982e108

   1. Initial program 4.1

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

   2. Simplified 6.3

      \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y \cdot z}} \]
      Proof
      (*.f64 x (/.f64 (sin.f64 y) (*.f64 y z))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (sin.f64 y) y) z))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)): 0 points increase in error, 2 points decrease in error

   3. Applied egg-rr 0.6

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

   if 9.99999999999999982e108 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 0.2

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

   2. Simplified 1.9

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\sin y} \cdot y}} \]
      Proof
      (*.f64 x (/.f64 (sin.f64 y) (*.f64 y z))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (sin.f64 y) y) z))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)): 0 points increase in error, 2 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -5 \cdot 10^{-199}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 10^{+109}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.005 \lor \neg \left(y \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}}\\ \end{array} \]

Alternative 2
Error	3.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-8} \lor \neg \left(y \leq 4 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+66} \lor \neg \left(z \leq 1.02 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 4
Error	2.9
Cost	6848

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

Alternative 5
Error	22.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -400:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{6}{z}}{y}\\ \mathbf{elif}\;y \leq 3.1:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -6}{y \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \lor \neg \left(y \leq 3.6 \cdot 10^{+63}\right):\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7
Error	22.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-6}{z}}{y \cdot y}\\ \end{array} \]

Alternative 8
Error	22.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-6}{y \cdot z}\\ \end{array} \]

Alternative 9
Error	22.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-6}{y \cdot z}\\ \end{array} \]

Alternative 10
Error	22.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \frac{6}{y}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-6}{y \cdot z}\\ \end{array} \]

Alternative 11
Error	22.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{6}{z}}{y}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-6}{y \cdot z}\\ \end{array} \]

Alternative 12
Error	22.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{6}{z}}{y}\\ \mathbf{elif}\;y \leq 250000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -6}{y \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 13
Error	22.7
Cost	704

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

Alternative 14
Error	25.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+34}:\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 15
Error	28.5
Cost	192

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

Reproduce

herbie shell --seed 2022340 
(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))