Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.9 → 0.3

Time: 7.3s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-315}:\\ \;\;\;\;\frac{x \cdot \sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ (sin y) y))))
   (if (<= t_0 -1e-298)
     (/ t_0 z)
     (if (<= t_0 4e-315)
       (/ (* x (sin y)) (* y z))
       (/ (/ 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 = x * (sin(y) / y);
	double tmp;
	if (t_0 <= -1e-298) {
		tmp = t_0 / z;
	} else if (t_0 <= 4e-315) {
		tmp = (x * sin(y)) / (y * z);
	} 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 = x * (sin(y) / y)
    if (t_0 <= (-1d-298)) then
        tmp = t_0 / z
    else if (t_0 <= 4d-315) then
        tmp = (x * sin(y)) / (y * z)
    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 = x * (Math.sin(y) / y);
	double tmp;
	if (t_0 <= -1e-298) {
		tmp = t_0 / z;
	} else if (t_0 <= 4e-315) {
		tmp = (x * Math.sin(y)) / (y * z);
	} 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 = x * (math.sin(y) / y)
	tmp = 0
	if t_0 <= -1e-298:
		tmp = t_0 / z
	elif t_0 <= 4e-315:
		tmp = (x * math.sin(y)) / (y * z)
	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(x * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= -1e-298)
		tmp = Float64(t_0 / z);
	elseif (t_0 <= 4e-315)
		tmp = Float64(Float64(x * sin(y)) / Float64(y * z));
	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 = x * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= -1e-298)
		tmp = t_0 / z;
	elseif (t_0 <= 4e-315)
		tmp = (x * sin(y)) / (y * z);
	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[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-298], N[(t$95$0 / z), $MachinePrecision], If[LessEqual[t$95$0, 4e-315], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]

Target

Original	2.9
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -9.99999999999999912e-299

   1. Initial program 0.2

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

   if -9.99999999999999912e-299 < (*.f64 x (/.f64 (sin.f64 y) y)) < 3.9999999989e-315

   1. Initial program 18.4

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

   2. Simplified 19.4

      \[\leadsto \color{blue}{\frac{\sin y}{z \cdot \frac{y}{x}}} \]
      Proof
      (/.f64 (sin.f64 y) (*.f64 z (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (sin.f64 y) (/.f64 y x)) z)): 15 points increase in error, 59 points decrease in error
      (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (sin.f64 y) y) x)) z): 10 points increase in error, 38 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (/.f64 (sin.f64 y) y))) z): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.5

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

   4. Taylor expanded in y around inf 0.8

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

   if 3.9999999989e-315 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y}{\sin y}}}}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 4 \cdot 10^{-315}:\\ \;\;\;\;\frac{x \cdot \sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.0
Cost	7112

\[\begin{array}{l} t_0 := \frac{x \cdot \sin y}{y \cdot z}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-47}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.3
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sin y}{\frac{z}{\frac{x}{y}}}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-38}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.9
Cost	6848

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

Alternative 4
Error	23.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -9.672631275697964 \cdot 10^{+83}:\\ \;\;\;\;0\\ \mathbf{elif}\;y \leq 3.55655911987175 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	47.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022295 
(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))