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

Average Error: 24.6 → 10.1

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(\frac{z \cdot x}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{y \cdot x}{a - z} + t \cdot \frac{1}{\frac{a - z}{z}}\right)\\ t_2 := \frac{y}{z} \cdot x + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+188}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{1}{a - z} \cdot \left(t - x\right), x\right)\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1
         (-
          (+ (/ (* z x) (- a z)) (+ x (/ (* y t) (- a z))))
          (+ (/ (* y x) (- a z)) (* t (/ 1.0 (/ (- a z) z))))))
        (t_2
         (+
          (* (/ y z) x)
          (+ t (- (* t (/ a z)) (+ (/ t (/ z y)) (* x (/ a z))))))))
   (if (<= z -1.5e+188)
     t_2
     (if (<= z -1.15e+69)
       (fma (- y z) (* (/ 1.0 (- a z)) (- t x)) x)
       (if (<= z -2.4e-190)
         t_1
         (if (<= z 9.6e-60)
           (+
            (* x (/ z (- a z)))
            (-
             (+ x (/ t (/ (- a z) y)))
             (+ (* z (/ t (- a z))) (* x (/ y (- a z))))))
           (if (<= z 2.15e+178) t_1 t_2)))))))

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 t_1 = (((z * x) / (a - z)) + (x + ((y * t) / (a - z)))) - (((y * x) / (a - z)) + (t * (1.0 / ((a - z) / z))));
	double t_2 = ((y / z) * x) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	double tmp;
	if (z <= -1.5e+188) {
		tmp = t_2;
	} else if (z <= -1.15e+69) {
		tmp = fma((y - z), ((1.0 / (a - z)) * (t - x)), x);
	} else if (z <= -2.4e-190) {
		tmp = t_1;
	} else if (z <= 9.6e-60) {
		tmp = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - ((z * (t / (a - z))) + (x * (y / (a - z)))));
	} else if (z <= 2.15e+178) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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)
	t_1 = Float64(Float64(Float64(Float64(z * x) / Float64(a - z)) + Float64(x + Float64(Float64(y * t) / Float64(a - z)))) - Float64(Float64(Float64(y * x) / Float64(a - z)) + Float64(t * Float64(1.0 / Float64(Float64(a - z) / z)))))
	t_2 = Float64(Float64(Float64(y / z) * x) + Float64(t + Float64(Float64(t * Float64(a / z)) - Float64(Float64(t / Float64(z / y)) + Float64(x * Float64(a / z))))))
	tmp = 0.0
	if (z <= -1.5e+188)
		tmp = t_2;
	elseif (z <= -1.15e+69)
		tmp = fma(Float64(y - z), Float64(Float64(1.0 / Float64(a - z)) * Float64(t - x)), x);
	elseif (z <= -2.4e-190)
		tmp = t_1;
	elseif (z <= 9.6e-60)
		tmp = Float64(Float64(x * Float64(z / Float64(a - z))) + Float64(Float64(x + Float64(t / Float64(Float64(a - z) / y))) - Float64(Float64(z * Float64(t / Float64(a - z))) + Float64(x * Float64(y / Float64(a - z))))));
	elseif (z <= 2.15e+178)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return 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_] := Block[{t$95$1 = N[(N[(N[(N[(z * x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y * x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(t * N[(1.0 / N[(N[(a - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] + N[(t + N[(N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision] - N[(N[(t / N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.5e+188], t$95$2, If[LessEqual[z, -1.15e+69], N[(N[(y - z), $MachinePrecision] * N[(N[(1.0 / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -2.4e-190], t$95$1, If[LessEqual[z, 9.6e-60], N[(N[(x * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + N[(t / N[(N[(a - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.15e+178], t$95$1, t$95$2]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.6
Target	12.0
Herbie	10.1

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

Derivation

1. Split input into 4 regimes

2. if z < -1.5e188 or 2.1500000000000001e178 < z

   1. Initial program 48.9

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

   2. Simplified 29.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   3. Taylor expanded in z around inf 23.1

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

   4. Simplified 6.8

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

   if -1.5e188 < z < -1.15000000000000008e69

   1. Initial program 33.6

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

   2. Simplified 18.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   3. Applied egg-rr 18.9

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

   if -1.15000000000000008e69 < z < -2.4e-190 or 9.60000000000000038e-60 < z < 2.1500000000000001e178

   1. Initial program 18.9

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

   2. Simplified 10.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   3. Taylor expanded in y around 0 14.9

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

   4. Applied egg-rr 12.7

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

   if -2.4e-190 < z < 9.60000000000000038e-60

   1. Initial program 8.0

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

   2. Simplified 7.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   3. Taylor expanded in y around 0 7.2

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

   4. Simplified 5.7

      \[\leadsto \color{blue}{\frac{z}{a - z} \cdot x + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(\frac{t}{a - z} \cdot z + \frac{y}{a - z} \cdot x\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{y}{z} \cdot x + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{1}{a - z} \cdot \left(t - x\right), x\right)\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-190}:\\ \;\;\;\;\left(\frac{z \cdot x}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{y \cdot x}{a - z} + t \cdot \frac{1}{\frac{a - z}{z}}\right)\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+178}:\\ \;\;\;\;\left(\frac{z \cdot x}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{y \cdot x}{a - z} + t \cdot \frac{1}{\frac{a - z}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))