Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.3 → 8.4

Time: 51.2s

Precision: binary64

Cost: 1096

Math

\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -8.507677451423992 \cdot 10^{+213}:\\ \;\;\;\;t - \frac{t - x}{\frac{z}{y - a}}\\ \mathbf{elif}\;z \leq 4.410018490600234 \cdot 10^{+178}:\\ \;\;\;\;x + \frac{y - z}{z - a} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{y - a}{z}\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.507677451423992e+213)
   (- t (/ (- t x) (/ z (- y a))))
   (if (<= z 4.410018490600234e+178)
     (+ x (* (/ (- y z) (- z a)) (- x t)))
     (+ t (* x (/ (- y a) z))))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.507677451423992e+213) {
		tmp = t - ((t - x) / (z / (y - a)));
	} else if (z <= 4.410018490600234e+178) {
		tmp = x + (((y - z) / (z - a)) * (x - t));
	} else {
		tmp = t + (x * ((y - a) / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - z) * (t - x)) / (a - z))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.507677451423992d+213)) then
        tmp = t - ((t - x) / (z / (y - a)))
    else if (z <= 4.410018490600234d+178) then
        tmp = x + (((y - z) / (z - a)) * (x - t))
    else
        tmp = t + (x * ((y - a) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * (t - x)) / (a - z));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.507677451423992e+213) {
		tmp = t - ((t - x) / (z / (y - a)));
	} else if (z <= 4.410018490600234e+178) {
		tmp = x + (((y - z) / (z - a)) * (x - t));
	} else {
		tmp = t + (x * ((y - a) / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return x + (((y - z) * (t - x)) / (a - z))

↓

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.507677451423992e+213:
		tmp = t - ((t - x) / (z / (y - a)))
	elif z <= 4.410018490600234e+178:
		tmp = x + (((y - z) / (z - a)) * (x - t))
	else:
		tmp = t + (x * ((y - a) / z))
	return tmp

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
end

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.507677451423992e+213)
		tmp = Float64(t - Float64(Float64(t - x) / Float64(z / Float64(y - a))));
	elseif (z <= 4.410018490600234e+178)
		tmp = Float64(x + Float64(Float64(Float64(y - z) / Float64(z - a)) * Float64(x - t)));
	else
		tmp = Float64(t + Float64(x * Float64(Float64(y - a) / z)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - z) * (t - x)) / (a - z));
end

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.507677451423992e+213)
		tmp = t - ((t - x) / (z / (y - a)));
	elseif (z <= 4.410018490600234e+178)
		tmp = x + (((y - z) / (z - a)) * (x - t));
	else
		tmp = t + (x * ((y - a) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.507677451423992e+213], N[(t - N[(N[(t - x), $MachinePrecision] / N[(z / N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.410018490600234e+178], N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(x * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	24.3
Target	11.7
Herbie	8.4

\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -8.5076774514239925e213

   1. Initial program 49.9

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf 23.8

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + t\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

   3. Simplified 6.8

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]
      Proof
      (-.f64 t (*.f64 (/.f64 (-.f64 t x) z) (-.f64 y a))): 0 points increase in error, 0 points decrease in error
      (-.f64 t (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 t x) (/.f64 z (-.f64 y a))))): 19 points increase in error, 33 points decrease in error
      (-.f64 t (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 t x) (-.f64 y a)) z))): 30 points increase in error, 26 points decrease in error
      (-.f64 t (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x)))) z)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 t (neg.f64 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 -1 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 y (-.f64 t x)) z) (/.f64 (*.f64 a (-.f64 t x)) z))))): 0 points increase in error, 1 points decrease in error
      (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z)) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z))) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z)) t)) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 5.7

      \[\leadsto t - \color{blue}{\frac{t - x}{\frac{z}{y - a}}} \]

   if -8.5076774514239925e213 < z < 4.41001849060023388e178

   1. Initial program 17.6

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Simplified 10.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{x - t}{z - a}, x\right)} \]
      Proof
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 x t) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 x t)))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 x t)))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 x) t))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 x)) t)) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 x)))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 t x))) (-.f64 z a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (neg.f64 (-.f64 t x)) (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (neg.f64 (-.f64 t x)) (-.f64 z a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (neg.f64 (-.f64 t x))) (*.f64 -1 (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (neg.f64 (-.f64 t x)))) (*.f64 -1 (-.f64 z a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (Rewrite=> remove-double-neg_binary64 (-.f64 t x)) (*.f64 -1 (-.f64 z a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z a)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) a))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) a)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= +-commutative_binary64 (+.f64 a (neg.f64 z)))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 (-.f64 y z) (/.f64 (-.f64 t x) (Rewrite<= sub-neg_binary64 (-.f64 a z))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))) x)): 19 points increase in error, 22 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) x): 77 points increase in error, 28 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around 0 16.0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot \left(y - z\right)}{z - a} + \left(\frac{\left(y - z\right) \cdot x}{z - a} + x\right)} \]

   4. Simplified 8.3

      \[\leadsto \color{blue}{x + \frac{y - z}{z - a} \cdot \left(x - t\right)} \]
      Proof
      (+.f64 x (*.f64 (/.f64 (-.f64 y z) (-.f64 z a)) (-.f64 x t))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 (-.f64 y z) (-.f64 z a)) x) (*.f64 (/.f64 (-.f64 y z) (-.f64 z a)) t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 y z) (/.f64 (-.f64 z a) x))) (*.f64 (/.f64 (-.f64 y z) (-.f64 z a)) t))): 11 points increase in error, 7 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a))) (*.f64 (/.f64 (-.f64 y z) (-.f64 z a)) t))): 44 points increase in error, 10 points decrease in error
      (+.f64 x (-.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (Rewrite<= *-commutative_binary64 (*.f64 t (/.f64 (-.f64 y z) (-.f64 z a)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (*.f64 t (Rewrite=> div-sub_binary64 (-.f64 (/.f64 y (-.f64 z a)) (/.f64 z (-.f64 z a))))))): 1 points increase in error, 2 points decrease in error
      (+.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (neg.f64 (*.f64 t (-.f64 (/.f64 y (-.f64 z a)) (/.f64 z (-.f64 z a)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (+.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 t (-.f64 (/.f64 y (-.f64 z a)) (/.f64 z (-.f64 z a)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (+.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (Rewrite=> mul-1-neg_binary64 (neg.f64 (*.f64 t (-.f64 (/.f64 y (-.f64 z a)) (/.f64 z (-.f64 z a)))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite=> unsub-neg_binary64 (-.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (*.f64 t (-.f64 (/.f64 y (-.f64 z a)) (/.f64 z (-.f64 z a))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (*.f64 t (Rewrite<= div-sub_binary64 (/.f64 (-.f64 y z) (-.f64 z a)))))): 2 points increase in error, 1 points decrease in error
      (+.f64 x (-.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 (-.f64 y z) (-.f64 z a)) t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 y z) t) (-.f64 z a))))): 36 points increase in error, 10 points decrease in error
      (+.f64 x (-.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t (-.f64 y z))) (-.f64 z a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (neg.f64 (/.f64 (*.f64 t (-.f64 y z)) (-.f64 z a)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (+.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 t (-.f64 y z)) (-.f64 z a)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a))) (*.f64 -1 (/.f64 (*.f64 t (-.f64 y z)) (-.f64 z a))))): 8 points increase in error, 13 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) x)) (*.f64 -1 (/.f64 (*.f64 t (-.f64 y z)) (-.f64 z a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 t (-.f64 y z)) (-.f64 z a))) (+.f64 (/.f64 (*.f64 (-.f64 y z) x) (-.f64 z a)) x))): 0 points increase in error, 0 points decrease in error

   if 4.41001849060023388e178 < z

   1. Initial program 50.1

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]

   2. Taylor expanded in z around inf 23.7

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + t\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

   3. Simplified 8.5

      \[\leadsto \color{blue}{t - \frac{t - x}{z} \cdot \left(y - a\right)} \]
      Proof
      (-.f64 t (*.f64 (/.f64 (-.f64 t x) z) (-.f64 y a))): 0 points increase in error, 0 points decrease in error
      (-.f64 t (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 t x) (/.f64 z (-.f64 y a))))): 19 points increase in error, 33 points decrease in error
      (-.f64 t (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 t x) (-.f64 y a)) z))): 30 points increase in error, 26 points decrease in error
      (-.f64 t (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x)))) z)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 t (neg.f64 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 y (-.f64 t x)) (*.f64 a (-.f64 t x))) z)))): 0 points increase in error, 0 points decrease in error
      (+.f64 t (*.f64 -1 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 y (-.f64 t x)) z) (/.f64 (*.f64 a (-.f64 t x)) z))))): 0 points increase in error, 1 points decrease in error
      (+.f64 t (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z)) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 t (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z))) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 y (-.f64 t x)) z)) t)) (*.f64 -1 (/.f64 (*.f64 a (-.f64 t x)) z))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 19.7

      \[\leadsto t - \color{blue}{-1 \cdot \frac{\left(y - a\right) \cdot x}{z}} \]

   5. Simplified 10.5

      \[\leadsto t - \color{blue}{x \cdot \frac{a - y}{z}} \]
      Proof
      (*.f64 x (/.f64 (-.f64 a y) z)): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite=> div-sub_binary64 (-.f64 (/.f64 a z) (/.f64 y z)))): 1 points increase in error, 1 points decrease in error
      (*.f64 x (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 a z) (neg.f64 (/.f64 y z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 a z)))) (neg.f64 (/.f64 y z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 (/.f64 a z)) (/.f64 y z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 y z) (neg.f64 (/.f64 a z)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 y z) (/.f64 a z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (neg.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 y a) z)))): 1 points increase in error, 1 points decrease in error
      (*.f64 x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (-.f64 y a) z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 (-.f64 y a) z) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 y a) (/.f64 z -1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (-.f64 y a)) (/.f64 z -1))): 35 points increase in error, 52 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 y a) x)) (/.f64 z -1)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (*.f64 (-.f64 y a) x) z) -1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -1 (/.f64 (*.f64 (-.f64 y a) x) z))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.507677451423992 \cdot 10^{+213}:\\ \;\;\;\;t - \frac{t - x}{\frac{z}{y - a}}\\ \mathbf{elif}\;z \leq 4.410018490600234 \cdot 10^{+178}:\\ \;\;\;\;x + \frac{y - z}{z - a} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{y - a}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	35.0
Cost	1768

\[\begin{array}{l} t_1 := x - t \cdot \frac{z}{a}\\ t_2 := \frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-301}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-252}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{x \cdot y}{z - a}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.020167982026052 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	32.1
Cost	1768

\[\begin{array}{l} t_1 := x + t \cdot \frac{y}{a}\\ t_2 := \frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-301}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-252}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{x \cdot y}{z - a}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 2.020167982026052 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	32.0
Cost	1768

\[\begin{array}{l} t_1 := x + t \cdot \frac{y}{a}\\ t_2 := \frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-301}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-252}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{x \cdot y}{z - a}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-24}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{t \cdot y}{a}\\ \mathbf{elif}\;a \leq 2.020167982026052 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.0
Cost	1360

\[\begin{array}{l} t_1 := t - \frac{t - x}{\frac{z}{y - a}}\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{-111}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t}}\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{elif}\;a \leq 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t - x}{a - z}}{\frac{1}{y}}\\ \end{array} \]

Alternative 5
Error	31.7
Cost	1240

\[\begin{array}{l} t_1 := x + t \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-301}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-252}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	28.7
Cost	1236

\[\begin{array}{l} t_1 := \frac{t \cdot \left(y - z\right)}{a - z}\\ t_2 := x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{if}\;a \leq -6 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-237}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{x \cdot y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	23.6
Cost	1236

\[\begin{array}{l} t_1 := x + \frac{y - z}{\frac{a}{t}}\\ t_2 := t + x \cdot \frac{y - a}{z}\\ \mathbf{if}\;a \leq -0.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	23.5
Cost	1236

\[\begin{array}{l} t_1 := t + x \cdot \frac{y - a}{z}\\ \mathbf{if}\;a \leq -0.55:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{elif}\;a \leq 2.020167982026052 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \end{array} \]

Alternative 9
Error	23.6
Cost	1236

\[\begin{array}{l} t_1 := t + x \cdot \frac{y - a}{z}\\ \mathbf{if}\;a \leq -0.55:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{elif}\;a \leq 2.020167982026052 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a}\\ \end{array} \]

Alternative 10
Error	19.2
Cost	1232

\[\begin{array}{l} t_1 := t - \frac{t - x}{\frac{z}{y - a}}\\ \mathbf{if}\;a \leq -6 \cdot 10^{-25}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{elif}\;a \leq 2.020167982026052 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a}\\ \end{array} \]

Alternative 11
Error	15.1
Cost	1232

\[\begin{array}{l} t_1 := t - \frac{t - x}{\frac{z}{y - a}}\\ t_2 := x + \frac{y - z}{\frac{a - z}{t}}\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.06 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	31.0
Cost	1108

\[\begin{array}{l} t_1 := \frac{t \cdot \left(y - z\right)}{a - z}\\ t_2 := x + t \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -6 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-237}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{x \cdot y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	24.7
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{if}\;a \leq -0.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;x - \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	34.8
Cost	976

\[\begin{array}{l} t_1 := x - t \cdot \frac{z}{a}\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.15 \cdot 10^{-301}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-252}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-100}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	34.4
Cost	712

\[\begin{array}{l} t_1 := x - t \cdot \frac{z}{a}\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-100}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	38.1
Cost	592

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6176168668280983 \cdot 10^{+213}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -2.851802674004011 \cdot 10^{+166}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-100}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	37.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{-100}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	45.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))